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Blog & Resources CalcDomain Blog & Resources Learn how to get more value from calculators: step‑by‑step guides, real‑world examples, and practical tips across finance, math, health, engineering, construction, and everyday life. Featured learning paths Master everyday finance decisions Use CalcDomain to plan a mortgage, compare loans, and grow your savings with compound interest. Mortgage Payment Calculator Compound Interest Calculator 50/30/20 Budget Calculator Study smarter with math & statistics Turn abstract formulas into interactive tools for algebra, geometry, and probability. Quadratic Formula Calculator Standard Deviation Calculator Pythagorean Theorem Calculator What is a blog, and what makes the CalcDomain blog different? A blog is a regularly updated collection of articles (called posts) that focus on specific topics and answer real questions from readers. Unlike static documentation, a blog evolves: new posts are added, old ones are updated, and content is organized so you can explore by topic or problem. The CalcDomain blog is focused on one thing: helping you solve real‑world problems with calculators. Instead of generic theory, each guide is built around: Concrete scenarios – e.g., “Should I refinance my mortgage?”, “How many calories do I burn on my run?”, “How much concrete do I need for this slab?” Interactive walkthroughs – every article links directly to the relevant calculators and shows sample inputs and outputs. Clear formulas – we show the math behind the tool, not just the final number. How to use this blog effectively Start from your question – “What do I want to decide or understand?” Find a guide – use the site search, categories, or the sections below (Finance, Math, Health, Engineering, Everyday Life). Follow the step‑by‑step examples – replicate them with your own numbers in the linked calculators. Save or export results – many calculators let you copy, print, or export key values for later reference. Explore by topic Finance & money Learn how to compare loans, plan retirement, optimize taxes, and evaluate investments using our finance calculators. Finance calculators overview Mortgage Payment Investment Return (ROI) Math, conversions & study help From unit conversions to advanced algebra and statistics, see how to turn formulas into quick, reliable answers. Math & Conversions hub Percentage Calculator Area Calculator Health & fitness Understand calories, macros, BMI, heart rate zones, and training metrics with calculators designed for clarity. Health & Fitness hub BMI Calculator Running Pace Engineering, science & construction Use professional‑grade tools for structural design, fluid flow, thermodynamics, and construction planning. Engineering calculators Construction & DIY Concrete Slab Volume Lifestyle & everyday tools Time and date helpers, text counters, cooking conversions, travel planning, and more everyday utilities. Lifestyle & Everyday hub Age Calculator Cooking Measurement Converter Example: from blog idea to calculator workflow Suppose you want to write (or follow) a guide on “How to decide if refinancing your mortgage makes sense.” A strong article would: Explain the key variables: current rate, new rate, remaining term, closing costs, and time you plan to stay in the home. Use the Mortgage Payment and Mortgage Refinance calculators to compare scenarios. Show the break‑even point: when interest savings exceed refinance costs. Discuss non‑numerical factors (job stability, risk tolerance, prepayment penalties). This is the pattern we follow across the CalcDomain blog: define the problem, choose the right calculators, walk through examples, then interpret the results. Blog & resources FAQ What is the CalcDomain blog? It’s a collection of guides, tutorials, and examples that show you how to use calculators to answer real questions in finance, math, health, engineering, construction, and everyday life. Is the content free to read? Yes. All articles and calculators on CalcDomain are free to access. No registration is required to run calculations or read guides. How often is the blog updated? We update content regularly based on new calculators, user feedback, and changes in topics like tax rules, financial products, and scientific methods. Can I suggest a topic or calculator? Yes. If there’s a calculation you perform often or a topic you’d like explained with examples, you can reach out via the Contact page. We prioritize topics that help many users solve recurring problems. Do blog posts replace professional advice? No. Our articles are educational and for general information only. For decisions involving your health, finances, taxes, or engineering projects, always consult a qualified professional and use the calculators as supporting tools. Related hubs & popular tools Lifestyle & Everyday Calculators Math & Conversions Finance Calculators Health & Fitness Engineering & Technical Quick links to useful calculators Mortgage Payment Calculator Percentage Calculator BMI Calculator Compound Interest Age Calculator Concrete Slab Volume

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CalcDomain Your trusted source for free online calculators. Accurate, fast, and reliable calculations for every need. Categories Finance Health & Fitness Math & Conversions Lifestyle & Everyday Construction & DIY Popular Tools Mortgage Calculator Percentage Calculator BMI Calculator Auto Loan Calculator House Affordability Support About Us Contact Privacy Policy Terms of Service Site Map © 2025 CalcDomain. All Rights Reserved. | Free Online Calculators for Everyone

Calculator Aggregate Type * Asphalt Content (%) * Calculate Results Stability 0 Data Source and Methodology All calculations are based on data from the authoritative manual: Mix Design PDF . All calculations are rigorously based on the formulas and data provided by this source. The Formula Explained \( \text{Stability} = \text{Function of Asphalt Content and Aggregate Type} \) Glossary of Variables Aggregate Type: The type of aggregate used in the mix. Asphalt Content (%): The percentage of asphalt in the mix. Stability: The resistance of the mixture to deformation. How It Works: A Step-by-Step Example To use this calculator, enter the aggregate type and asphalt content. The calculator will then use these inputs to calculate the stability of the mix using the Marshall method. Frequently Asked Questions (FAQ) What is the Marshall Mix Design? The Marshall Mix Design is a procedure to design and evaluate asphalt mixtures and is widely used in road construction. What inputs are needed for the calculation? The calculator requires inputs like aggregate type and asphalt content to compute the stability of the asphalt mix. Why is asphalt content important? Asphalt content affects the durability and stability of the pavement. Proper content ensures a balance between flexibility and strength. Can I use this calculator for all types of asphalt mixes? While this calculator provides a general guideline, specific projects may require additional analysis and expert evaluation. How accurate are the results? The results are based on the standard Marshall method, providing reliable estimates, but should be validated by professional engineers.

Calorie Calculator All required fields are marked with * . Units Metric (kg, cm) Imperial (lb, ft/in) Sex * Select… Male Female ? Used in Mifflin–St Jeor to adjust BMR constant. Age (years) * ? Typical valid range 5–100 years. Height (cm) * ? Enter height in centimeters (e.g., 175.0). Height (ft/in) * Weight (kg) * ? Enter body weight in kilograms. Weight (lb) * Body Fat % (optional) ? Enables Katch–McArdle (uses lean mass). Leave blank to use Mifflin–St Jeor. Formula Auto (Katch if body-fat given, else Mifflin) Mifflin–St Jeor Katch–McArdle Choose explicitly if you prefer a specific method. Activity level (PAL) * Select… Sedentary (PAL 1.2) Lightly active (PAL 1.375) Moderately active (PAL 1.55) Very active (PAL 1.725) Extra active (PAL 1.9) ? Physical Activity Level multipliers approximate daily energy expenditure. Goal Maintain Lose weight Gain weight Target rate (kg/week) Used to estimate daily deficit/surplus (≈ 7,700 kcal/kg). Protein (g/kg) Common ranges 1.2–2.2 g/kg depending on goals. Fat (% of calories) Remaining calories go to carbs after protein & fat. Calculate Reset Results BMR — — TDEE — kcal/day Goal calories — — Daily macros Protein — Fat — Carbs — Fiber (guide) — Macro breakdown Data source & methodology Basal Metabolic Rate : Mifflin MD, St Jeor ST et al. (1990), “A new predictive equation for resting energy expenditure in healthy individuals.” Am J Clin Nutr 51(2):241-247. Lean-mass BMR : Katch VL, McArdle WD (1996), “Nutrition, Weight Control, and Exercise.” 6th ed. (Katch–McArdle formula). Activity factors (PAL) : Derived from widely used Physical Activity Level categories as summarized in IOM/WHO guidance. Tutti i calcoli si basano rigorosamente sulle formule e sui dati forniti da questa fonte. The formula explained \[ \textbf{Mifflin–St Jeor:}\quad \mathrm{BMR}_{\text{male}} = 10W + 6.25H - 5A + 5,\qquad \mathrm{BMR}_{\text{female}} = 10W + 6.25H - 5A - 161 \] \[ \textbf{Katch–McArdle:}\quad \mathrm{BMR} = 370 + 21.6 \times LBM,\quad LBM = W \times (1 - \frac{\%BF}{100}) \] \[ \textbf{Total Daily Energy Expenditure:}\quad \mathrm{TDEE} = \mathrm{BMR} \times \mathrm{PAL} \] \[ \textbf{Goal Calories:}\quad \mathrm{kcal}_{\text{goal}} \approx \mathrm{TDEE} + \frac{7700 \times \Delta \mathrm{kg/week}}{7} \] Glossary of variables Symbol Meaning Units W Body weight kg H Height cm A Age years %BF Body fat percentage % LBM Lean body mass kg PAL Activity multiplier dimensionless How it works: a step-by-step example Inputs: Sex = female; Age = 32; Height = 168 cm; Weight = 62 kg; Body fat blank (use Mifflin); Activity = 1.55; Goal = lose 0.3 kg/week; Protein = 1.6 g/kg; Fat = 30%. Mifflin BMR = 10×62 + 6.25×168 − 5×32 − 161 = 620 + 1050 − 160 − 161 = 1,349 kcal . TDEE = 1,349 × 1.55 = 2,091 kcal . Daily adjustment ≈ 7,700×0.3/7 ≈ 330 kcal deficit → Goal calories ≈ 1,761 kcal . Protein = 1.6×62 = 99 g → 396 kcal. Fat = 30% of 1,761 = 528 kcal → 59 g. Carbs = remaining ≈ 1,761 − (396+528) = 837 kcal → 209 g. Frequently asked questions Which formula should I choose? Mifflin–St Jeor fits most people well. If you have a reliable body-fat % (e.g., DEXA or consistent calipers), Katch–McArdle personalises BMR via lean mass. How do I pick my activity level? Choose a PAL that reflects an average week. Start conservative; adjust calories based on actual weight trend over 2–3 weeks. Is 7,700 kcal per kg exact? It’s an approximation for adipose tissue energy. Real-world weight change also involves water and glycogen shifts. What macro split is best? Prioritise protein (1.2–2.2 g/kg). Keep fat ≥ 20–25% for hormones and absorption. Allocate remaining calories to carbs based on preference and training. How often should I recalc? Every 4–8 weeks or when body weight/activity changes meaningfully. Tool developed by Ugo Candido. Content verified by CalcDomain Editorial Board. Last accuracy review: October 30, 2025 Why this tool is different Dual formulae (Mifflin/Katch) with body-fat support. Goal calories from target weekly change. Macro grams with a clear, printable summary. WCAG 2.1 AA, keyboard-first, screen-reader friendly. Ad Placement (300×600)

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Covariance Calculator This calculator helps investors and analysts measure the covariance between two assets in a portfolio, which is crucial for risk management and diversification strategies. Calculate Covariance Asset 1 Returns (%) * Asset 2 Returns (%) * Calculate Covariance 0.00 Data Source and Methodology All calculations are based on standard financial algorithms for covariance. For detailed information, refer to Investopedia . All calculations adhere strictly to provided formulas and data. The Formula Explained Cov(X, Y) = E[(X - E[X])(Y - E[Y])] Glossary of Variables Covariance (Cov): A measure of how two assets move together. Asset 1 Returns: The percentage return of the first asset. Asset 2 Returns: The percentage return of the second asset. E[X]: Expected value of X (mean of asset 1 returns). E[Y]: Expected value of Y (mean of asset 2 returns). How It Works: A Step-by-Step Example Consider two assets with returns over a period. By entering their returns in the calculator, the covariance is computed as the average of the products of their deviations from their respective means. Frequently Asked Questions (FAQ) What is covariance in finance? Covariance is a measure of the relationship between two variables in a financial context. It indicates how two asset returns move together. How do I calculate covariance? Covariance is calculated by finding the average of the product of the deviations of each pair of data points in two datasets. Our calculator simplifies this process. Why is covariance important? Covariance is crucial in portfolio management as it helps in diversification by understanding how different assets correlate. Can covariance be negative? Yes, a negative covariance indicates that as one asset's return increases, the other's decreases, which can be useful for diversification. Is a higher covariance better? Not necessarily. Higher covariance indicates stronger correlation, but for diversification, low or negative covariance is often preferred. Tool developed by Ugo Candido . Content reviewed by the CalcDomain Expert Team . Last reviewed for accuracy on: October 1, 2023.

Disclaimer Calculator Disclaimer Important: Please read this disclaimer carefully before using any CalcDomain calculators. General Disclaimer The calculators provided by CalcDomain are for informational and educational purposes only . They do not constitute financial, tax, investment, legal, engineering, medical, or other professional advice and should not be used as a substitute for consultation with qualified professionals. Professional Consultation Required For important decisions involving any of the following, consult a qualified professional and verify results independently: Financial planning & accounting: Certified financial professionals or accountants Tax calculations & filings: Licensed tax professionals or official tax authorities Investments: Licensed investment advisors Legal matters: Qualified legal counsel Engineering & safety-critical design: Licensed professional engineers Health & fitness: Qualified medical or health professionals Accuracy and Limitations We work to keep our tools accurate and up to date; however: Results are approximations and may omit variables relevant to your specific situation. Regulations, codes, standards, rates, and formulas can change without notice. Inputs provided by users may be incomplete, imprecise, or incorrect. Rounding, unit conversions, and edge-case behaviors can affect outcomes. No Warranty CalcDomain provides the calculators and related content “as is” and “as available” , without warranties of any kind, express or implied, including but not limited to accuracy, completeness, reliability, merchantability, fitness for a particular purpose, or non-infringement. Limitation of Liability To the maximum extent permitted by law, CalcDomain and its contributors shall not be liable for any direct, indirect, incidental, special, consequential, exemplary, or punitive damages arising from or related to your use of the calculators or reliance on their results, including (without limitation): Financial or investment losses Business or operational decisions Tax or regulatory compliance issues Engineering, safety, or design outcomes Health or medical outcomes User Responsibility By using our calculators, you agree that you are responsible for: Providing accurate and complete inputs Understanding the scope and limitations of each calculator Verifying results with appropriate professionals and official sources Ensuring compliance with local laws, codes, and regulations Using the tools only for their intended purposes Formula Sources Calculators are based on widely accepted formulas, standards, or published methods. Requirements may vary by jurisdiction, industry, or application. Always confirm that the methods used are appropriate for your specific needs. External Links and References Any external links or references are provided for convenience and do not imply endorsement. CalcDomain is not responsible for the content, accuracy, or availability of external sites. No Professional–Client Relationship Your use of CalcDomain does not create a professional–client relationship (financial, legal, engineering, medical, or otherwise) between you and CalcDomain or its contributors. Updates and Changes We may update calculators, methods, and this disclaimer at any time. Please review this page periodically to ensure you are using the most current information. Last updated: 2025-01-01 Remember: These calculators support preliminary analysis and learning. For any critical, official, or safety-related purposes, consult qualified professionals and verify results independently. By using CalcDomain, you also agree to our Terms and Privacy Policy .

Discover how your biorhythms affect your daily life. This calculator helps you understand your physical, emotional, and intellectual cycles based on your birth date. Calculate Your Biorhythms Enter your Birth Date: Calculate Results Physical Cycle: - Emotional Cycle: - Intellectual Cycle: - Data Source and Methodology All calculations are based on established biorhythm theories. For more information, visit our source . All calculations are strictly based on the formulas and data provided by this source. The Formula Explained The biorhythm calculations are based on sinusoidal functions. The formulas used are: Physical Cycle: \( \sin\left(\frac{2\pi \times \text{days}}{23}\right) \) Emotional Cycle: \( \sin\left(\frac{2\pi \times \text{days}}{28}\right) \) Intellectual Cycle: \( \sin\left(\frac{2\pi \times \text{days}}{33}\right) \) Glossary of Variables Birth Date: The date when you were born. Physical Cycle: Reflects your physical strength, health, and endurance. Emotional Cycle: Reflects your emotional stability and sensitivity. Intellectual Cycle: Reflects your mental capacity and alertness. How It Works: A Step-by-Step Example Consider a person born on January 1, 2000. If today's date is January 1, 2023, they are 23 years old. The calculation for today's physical cycle would be: Physical Cycle: \( \sin\left(\frac{2\pi \times 8395}{23}\right) \) Frequently Asked Questions (FAQ) What are biorhythms? Biorhythms are cycles that are believed to affect physical, emotional, and intellectual aspects of an individual. How often do these cycles occur? The physical cycle is 23 days, the emotional cycle is 28 days, and the intellectual cycle is 33 days. Do biorhythms really affect us? While some believe in their influence, scientific evidence is mixed. They can offer a unique perspective on personal rhythms. Can I calculate biorhythms for any date? Yes, you can calculate biorhythms for any given date using your birth date as a reference. Is this calculator accurate? The calculator uses standard biorhythm formulas. Accuracy depends on the input data and the theoretical basis of biorhythms. Tool developed by Ugo Candido . Content verified by experts. Last reviewed on October 5, 2023.

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Help Center Welcome to the CalcDomain Help Center. Find answers to common questions and learn how to get the most out of our professional calculators. Getting Started How to Use Our Calculators Step-by-step guides for each calculator type, including tips for accurate results and understanding the formulas. Understanding Formulas Learn about the mathematical formulas behind our calculations and how they comply with industry standards. Export & Sharing How to export your calculation results to PDF, print them, or share with colleagues. Frequently Asked Questions Are the calculators accurate? Yes, all our calculators use industry-standard formulas and are validated against regulatory guidelines. Each calculator shows the exact formulas used for complete transparency. Can I use these calculators offline? Yes! CalcDomain is a Progressive Web App (PWA) that works offline once loaded. You can even install it on your device. Is CalcDomain free to use? Absolutely. All calculators are completely free with no registration required. We believe professional tools should be accessible to everyone. How do I report a bug or suggest improvements? Please use our contact form to report bugs or suggest new features. We value your feedback! Need More Help? If you can’t find what you’re looking for, please contact our support team . We typically respond within 24 hours.

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Home › Advanced Math & Statistics › Modular inverse Modular inverse calculator This modular inverse calculator finds the modular multiplicative inverse of an integer \(a\) modulo \(m\). In other words, it finds an integer \(x\) such that \(a x \equiv 1 \pmod m\), or tells you when no such inverse exists. The tool is built for students, engineers and cryptography practitioners who need a reliable, step-by-step implementation of the extended Euclidean algorithm. Integer \(a\) Any non-zero integer. The calculator will automatically reduce it modulo \(m\). Modulus \(m\) Positive integer > 1. For cryptography, \(m\) is often a large prime or product of primes. Show extended Euclidean algorithm steps Compute modular inverse Reset Quick examples Click an example to auto-fill the fields and see a typical modular inverse computation. a = 3, m = 11 a = 17, m = 43 a = 35, m = 64 a = −7, m = 26 Result Extended Euclidean algorithm steps The table below shows the recurrence for the extended Euclidean algorithm applied to the reduced value \(a' = a \bmod m\) and the modulus \(m\). Step q r s t What is a modular inverse? Given an integer \(a\) and a modulus \(m > 1\), a modular multiplicative inverse of \(a\) modulo \(m\) is an integer \(x\) such that \[ a x \equiv 1 \pmod m. \] Intuitively, \(x\) plays the role of a reciprocal of \(a\) in modular arithmetic: multiplying by \(x\) “undoes” the effect of multiplying by \(a\), but only up to congruence modulo \(m\). Existence and uniqueness A modular inverse exists if and only if \(\gcd(a, m) = 1\) (that is, \(a\) and \(m\) are coprime). If an inverse exists, it is unique modulo \(m\) . All solutions differ by a multiple of \(m\). If \(\gcd(a, m) > 1\), then no modular inverse exists. Algorithm used by this calculator This calculator uses the extended Euclidean algorithm , a classic number-theoretic method that not only computes \(\gcd(a, m)\) but also finds integers \(x\) and \(y\) such that \[ a x + m y = \gcd(a, m). \] When \(\gcd(a, m) = 1\), the coefficient \(x\) is precisely a modular inverse of \(a\) modulo \(m\). We then reduce \(x\) to the standard range \(0 \le x \le m-1\). Extended Euclidean algorithm (high-level outline) Reduce the input \(a\) modulo \(m\): set \(a' = a \bmod m\). Initialize \(r_0 = a'\), \(r_1 = m\), \(s_0 = 1\), \(s_1 = 0\), \(t_0 = 0\), \(t_1 = 1\). Repeat until \(r_k = 0\): Compute the quotient \(q_k = \lfloor r_{k-1}

Home › Core Math & Algebra › Spearman's Rank Correlation Spearman's Rank Correlation Calculator (ρ) Paste your paired data, compute Spearman's rank correlation coefficient ρ, and see the full ranking table with d and d². Non-parametric stats 1. Enter paired data (X, Y) Enter each pair on a new line, separated by a space, comma, or tab. Example: 10 12 . Paired observations Load example data Clear Minimum 3 pairs are required. The calculator assigns average ranks when there are ties. 2. Options Show full ranking table (ranks, d, d²) Also show classic Σd² formula (no ties) Calculate Spearman's ρ 3. Results Spearman's coefficient ρ – Sample size n: – Strength & direction: ρ ranges from -1 (perfect decreasing monotonic relationship) to +1 (perfect increasing monotonic relationship). Test statistic (approximate) t statistic: – Degrees of freedom: – For moderate to large n, you can approximate a significance test using a t distribution with n − 2 degrees of freedom. This calculator does not compute p-values; use statistical software for formal hypothesis tests. Classic Σd² formula (valid when there are no ties) Σd² = – , n = – . If there are no ties, ρ can also be computed as ρ = 1 − 6 Σd²

Home › Lifestyle & Everyday › General Utility › CIDR Calculator CIDR Calculator (IPv4) Convert CIDR to IP ranges, subnet and supernet networks, summarize multiple CIDR blocks, and detect overlaps. Built for network engineers, SREs, and students. 1. Single CIDR

Home › Math & Conversions › Core Math & Algebra › Bisection Method Calculator Bisection Method Calculator Use the bisection method to numerically approximate a root of \( f(x) = 0 \) on an interval \([a, b]\). This tool checks the sign-change condition, runs the iterations step-by-step, and gives you the final approximation together with an iteration table and theoretical iteration bound. Ideal for numerical analysis courses, engineering calculations and quick checks when you need a guaranteed-convergence root-finding method. 1. Enter function and interval Function \( f(x) \) Use x as the variable. Supported functions include sin , cos , tan , exp , log (natural log), sqrt , and more via JavaScript’s Math object. Use ^ or ** for powers. Left endpoint \( a \) Right endpoint \( b \) The bisection method requires opposite signs at the endpoints: \( f(a) \cdot f(b) < 0 \). 2. Set tolerance and iteration limit Tolerance \( \varepsilon \) Target accuracy for the root and interval length. Max iterations Safety cutoff to prevent infinite loops. Decimals in output Display precision in the table and results. Calculate root Clear table Deterministic & guaranteed convergence (with sign change) Bisection method results Root approximation Approximate root f(root) Final interval Half-length Iterations used Max iterations Theoretical iteration bound This bound assumes only the interval-length criterion \( \frac{b-a}{2^N} \le \varepsilon \). The actual stopping rule also checks \( |f(m)| \le \varepsilon \). Iteration table Copy as CSV # a b m f(a) f(m) f(b) (b − a)/2 Each row shows the interval \([a_k, b_k]\), its midpoint \( m_k \), the function values at the endpoints and midpoint, and the half-length of the interval. The method always keeps a sign change across the current interval. The bisection method – theory in a nutshell The bisection method is one of the simplest and most robust algorithms for solving nonlinear equations of the form \( f(x) = 0 \). It is based on the Intermediate Value Theorem : if \( f \) is continuous on \([a, b]\) and \( f(a) \) and \( f(b) \) have opposite signs, then there exists at least one root \( \xi \in (a, b) \) such that \( f(\xi) = 0 \). Bisection algorithm Check that \( f(a) \cdot f(b) < 0 \). If not, the method cannot start. Compute the midpoint \( m = \dfrac{a + b}{2} \). Evaluate \( f(m) \). If \( f(m) = 0 \) (or \( |f(m)| \) is within tolerance), stop: \( m \) is the root approximation. Otherwise, choose the new interval: If \( f(a) \cdot f(m) < 0 \), set \( b \gets m \). Else, set \( a \gets m \). Repeat from step 2 until the interval is smaller than the tolerance or you hit the iteration limit. Convergence and error bound Let the initial interval length be \( L_0 = b_0 - a_0 \). After \( N \) bisection steps, the interval length is \[ L_N = \frac{L_0}{2^N}. \] To guarantee that the interval length is at most \( \varepsilon \), we need \[ \frac{L_0}{2^N} \le \varepsilon \quad \Rightarrow \quad N \ge \log_2 \left( \frac{L_0}{\varepsilon} \right). \] This gives a simple a priori bound on the number of iterations needed. The method converges linearly to the root. Advantages and limitations Advantages Guaranteed convergence when \( f \) is continuous and there is a sign change on \([a, b]\). Very stable and simple to implement. No derivatives required. Limitations Requires an interval with a sign change (you must bracket a root). Converges slower than methods like Newton–Raphson or secant. Only finds one root per bracketing interval (others require separate intervals). Bisection method – FAQ What happens if f(a) and f(b) have the same sign? + If \( f(a) \) and \( f(b) \) do not have opposite signs, the bisection method cannot guarantee a root between a and b. This calculator will show an error and refuse to start. In practice, you should scan the function or plot it to locate intervals where the sign changes. How does this compare to Newton’s method? + Newton’s method typically converges much faster (quadratically near the root) but requires a good initial guess and the derivative \( f'(x) \). It can diverge if the guess is poor. The bisection method is slower but much more robust, provided you can bracket a root with a sign change. What tolerance should I choose? + It depends on your application. For teaching examples and quick checks, \( 10^{-4} \) or \( 10^{-6} \) is usually enough. For high-precision scientific work you may want \( 10^{-8} \) or smaller, but keep in mind machine precision limits and that the function itself might not be known that accurately. Can this method find complex roots? + No. The classic bisection method works on real-valued functions on real intervals. To locate complex roots you need different techniques (for example, complex Newton methods or contour integral methods). Core Math & Algebra tools G-test (likelihood-ratio test) FOIL Method Calculator Graph Adjacency Matrix Speed, Distance, Time Friedman Test Traveling Salesman Problem (TSP) Stem and Leaf Plot Generator Bisection Method – you are here Monte Carlo Simulation Lottery Odds Dot Product Matrix Determinant Partial Fraction Decomposition Trapezoid Area Combination and Permutation More numerical methods Newton–Raphson Method Secant Method Euler’s Method (ODE) Taylor Series Expansion

Home › Math & Conversions › Core Math & Algebra › Chinese Remainder Theorem Calculator Chinese Remainder Theorem Calculator Solve systems of congruences of the form \(x \equiv a_i \pmod{m_i}\). The tool checks consistency, handles non-coprime moduli, and shows a least nonnegative solution when it exists. Designed for number theory, cryptography, and contest training where a transparent, step-by-step CRT solver is more useful than a black box. CRT system solver Generalized Chinese remainder theorem Enter your congruences \(x \equiv a_i \pmod{m_i}\) and click Calculate . The solver uses the extended Euclidean algorithm, detects incompatibilities, and reports the solution modulo the combined modulus. Number of congruences Between 2 and 8 rows. Empty rows are ignored. Update table Load classic example Example: \(x \equiv 2 \pmod{3}\), \(x \equiv 3 \pmod{5}\), \(x \equiv 2 \pmod{7}\). Notes: Moduli \(m_i\) must be positive integers. The solver uses arbitrary-precision integers (BigInt) for robustness. Calculate Clear Backend: extended Euclidean algorithm and iterative CRT merging. Solution, combined modulus, and step-by-step CRT construction will appear here after you click Calculate . Chinese remainder theorem – definition and intuition The Chinese remainder theorem (CRT) describes when a system of congruences \[ x \equiv a_1 \pmod{m_1},\quad x \equiv a_2 \pmod{m_2},\quad \dots,\quad x \equiv a_k \pmod{m_k} \] has a solution, and how to describe all such solutions. Informally: if the congruences are mutually compatible, there is a unique solution modulo the least common multiple of the moduli. Classical CRT with pairwise coprime moduli In the classical setting, the moduli \(m_1,\dots,m_k\) are pairwise coprime \((\gcd(m_i,m_j) = 1\) for \(i \neq j)\). Let \(M = m_1 m_2 \cdots m_k\) and \[ M_i = \frac{M}{m_i},\qquad N_i \equiv M_i^{-1} \pmod{m_i}, \] where \(M_i^{-1}\) denotes the modular inverse of \(M_i\) modulo \(m_i\). Then one explicit solution is \[ x_0 \equiv \sum_{i=1}^k a_i M_i N_i \pmod{M}. \] Every solution is congruent to \(x_0\) modulo \(M\), so the full solution set is \(x = x_0 + tM\) for \(t \in \mathbb{Z}\). Generalized CRT with non-coprime moduli When the moduli are not pairwise coprime, the situation is more subtle. For two congruences \[ x \equiv a_1 \pmod{m_1},\qquad x \equiv a_2 \pmod{m_2}, \] a solution exists if and only if \(\;a_1 \equiv a_2 \pmod{\gcd(m_1, m_2)}\). If this compatibility condition holds, the two congruences have a common solution, and the combined modulus is \(\operatorname{lcm}(m_1, m_2)\). The calculator implements this generalized CRT by merging congruences one pair at a time, always checking the relevant gcd condition. If any pair is incompatible, it reports that the system has no solution. Algorithm used by the calculator Let \(x \equiv a \pmod{n}\) describe the current combined congruence, and suppose we want to merge \(x \equiv b \pmod{m}\). Compute \(g = \gcd(n, m)\). If \(a \not\equiv b \pmod{g}\), the system is inconsistent and has no solution. Otherwise set \(n_1 = n/g\), \(m_1 = m/g\), and \(r = (b - a)/g\). Find the modular inverse of \(n_1\) modulo \(m_1\) using the extended Euclidean algorithm. Compute \(t \equiv r \cdot n_1^{-1} \pmod{m_1}\) and set \(x = a + n t\), \(N = n \cdot m_1\). After processing every congruence, we end up with a single congruence \(x \equiv x_0 \pmod{N}\) that encodes all solutions, where \(N\) is the least common multiple of the original moduli. Applications: from calendars to cryptography The Chinese remainder theorem appears in many contexts: calendar calculations and periodic scheduling, coding theory and cryptographic schemes such as RSA, modular arithmetic with large integers via residue systems, programming contests and algorithmic number theory. In each case, CRT allows a large modular problem to be broken into smaller, independent pieces that are easier to work with and then recombined in a mathematically sound way. Chinese remainder theorem – FAQ What happens if some moduli share a common factor? + The calculator does not require the moduli to be pairwise coprime. If two moduli share a non-trivial gcd, it checks whether the corresponding remainders agree modulo that gcd. If they do, the congruences can still be merged; otherwise, the system has no solution and the conflicting pair is reported explicitly. Why is the solution unique modulo the combined modulus? + Once a solution x₀ exists, any other integer x that satisfies all congruences must differ from x₀ by a multiple of every modulus, hence by a multiple of their least common multiple M. Conversely, x₀ + kM satisfies each congruence for every integer k, so the solution set is exactly one residue class modulo M. Can I use this tool for cryptographic-size numbers? + The implementation uses JavaScript BigInt, which supports very large integers. However, browsers are not optimized for extremely heavy big-integer workloads. For production cryptographic deployments you should rely on vetted, specialized libraries and carefully audited code rather than a general-purpose web calculator. Core Math & Algebra tools Modular Arithmetic Equation Solver Circular Segment Area Trig Identity Explorer Hyperbolic Function Explorer Linear Algebra Toolbox Trapezoidal Rule Calculator Perfect Number Calculator Chinese Remainder Theorem (you are here) Set Theory Basics Number theory & discrete math Base Converter Interval Notation Helper Pythagorean Theorem Convolution Explorer Dice Roll Probability Professional use note This CRT calculator is ideal for teaching, exam preparation, and prototyping of modular arithmetic schemes. For production cryptography or safety-critical applications, validate all logic with independent implementations and domain-specific tooling.

Home › Math & Conversions › Core Math & Algebra › Circular Segment Area Circular Segment Area Calculator Compute the area of a circular segment from radius and height, radius and central angle, or radius and chord length. The tool also reports chord length, arc length, sector area, and the segment’s share of the full circle. Input mode Radius and segment height (sagitta) Radius and central angle Radius and chord length Choose the quantities you know. All length inputs must use the same unit (e.g., meters, centimeters). Radius R Circle radius (R > 0). Segment height h Sagitta: distance from the chord up to the circle. Must satisfy 0 < h < 2R. Central angle θ Units: degrees radians Use 0 < θ < 360° (or 0 < θ < 2π radians) for a proper segment. Chord length c Straight-line distance between the chord endpoints. Must satisfy 0 < c < 2R. Calculate Clear Enter the known values and click Calculate to see segment area, chord and arc length, sector area, and more. What is a circular segment? A circular segment is the region of a circle cut off by a chord. It is bounded by: a straight line (the chord), and the corresponding arc of the circle. If you draw a chord across a circle and shade only the “cap” between the chord and the circle, you get a circular segment. In many applications (fluid levels in tanks, structural design, optics, surveying) we know the radius and one extra measure such as the segment height, chord length, or central angle and need the segment area. Key geometry and notation We will use the following notation: R – circle radius h – segment height (sagitta), measured from the chord to the circle along the symmetry axis c – chord length θ – central angle corresponding to the segment (in radians) A_seg – circular segment area A_sec – circular sector area Formulas for circular segment area The fundamental formula for the area of a circular segment in terms of radius and central angle (in radians) is: Segment area from radius and central angle Given a circle of radius R and central angle θ (radians), A_sec = (1/2) · R² · θ (sector area) A_tri = (1/2) · R² · sin θ (area of isosceles triangle formed by the two radii and the chord) A_seg = A_sec − A_tri = (R²

Home › Math & Conversions › Core Math & Algebra › Cohen’s d Calculator Cohen’s d Calculator (Effect Size) This Cohen’s d calculator computes standardized effect sizes for the difference between means. It supports independent samples , one-sample , and paired

Home › Math & Conversions › Core Math & Algebra › Continued fraction Continued Fraction Calculator Convert decimals and rational numbers to simple continued fractions, compute convergents, and see best rational approximations step-by-step. Core Math & Algebra Interactive continued fraction expansion and convergents Choose how you want to define your number and what you want to compute. The tool supports: Decimal or rational input \((p/q)\) with controlled depth and tolerance. Simple continued fraction expansion \([a_0; a_1, \dots, a_n]\). Convergents table with \((p_k, q_k)\), decimal values, and errors. Reverse conversion from a continued fraction back to a number. Decimal

Home › Math & Conversions › Core Math & Algebra › Convolution Convolution Calculator Compute discrete convolution of two sequences, inspect each sum in \(y[k] = \sum_n x[n]\,h[k-n]\), and see how convolution behaves in signals, probability and deep learning. Core Math & Algebra Interactive discrete convolution of finite sequences Enter two finite sequences (signals, kernels, filters, probability mass functions). The tool will: Parse sequences like 1, 2, 3 or 0.5 -1 0.5 . Compute full , same , or valid convolution. Show each output sample as a sum of products. Provide quick statistics (length, energy) and notes on interpretation. Discrete convolution Notes & examples Input sequence \(x[n]\) Separate values with commas or spaces. Up to ~50 samples for each sequence. Kernel

Home › Math & Conversions › Core Math & Algebra › Dice roll probability Dice Roll Probability Calculator Compute the probability of sums and successes when rolling one or many dice. Supports exact, at least, at most, and shows odds, percentages, and step-by-step explanations. Core Math & Algebra Interactive dice probability engine Choose whether you care about the sum of the dice or the number of successes (for example “roll 5+ on each die”), then enter your scenario. Sum of dice Number of successes Number of dice 1–15 recommended for exact sums. Sides per die 6 for a standard d6, 20 for a d20, etc. Decimal places 2 decimals 4 decimals 6 decimals Sum condition Type Exactly equal to At least At most Target sum Between min = dice × 1 and max = dice × sides. Show full sum distribution (for small dice pools) Success condition “Success” faces per die For a d6, success on 6 ⇒ 1 face; on 5 or 6 ⇒ 2 faces. Number of successes Exactly At least At most Target successes (k) For “at least one success” set k = 1 and type = “At least”. Calculate probability Clear Example: 2d6 sum = 7 Example: at least one 6 on 4d6 Results Main probability Details Odds & checks Sum distribution Distribution is exact for fair dice. For large pools, only the requested probability is shown to keep things fast. Dice roll probability in a nutshell When you roll fair dice, each face is equally likely. For one s-sided die, the probability of any particular face is \(\dfrac{1}{s}\). For multiple dice, the number of possible outcomes grows quickly: rolling \(n\) independent s-sided dice gives \(s^n\) equally likely outcomes. To get the probability of an event (such as “sum equals 7” or “at least one 6”), you count how many outcomes make the event happen and divide by \(s^n\). Sum of n dice: exact probabilities For \(n\) fair s-sided dice, the sum \(S\) can range from \(n\) (all 1s) up to \(n\,s\) (all s). The probability that the sum equals a specific value \(t\) is \(\mathbb{P}(S = t) = \dfrac{\text{number of combinations of faces that add to } t}{s^n}\). For small \(n\), you can count by hand or via a dynamic programming table that accumulates the number of ways to reach each sum as you add dice one by one. This is exactly what the calculator does internally. Example: 2d6 sum of 7 Two standard six-sided dice (2d6) have \(6^2 = 36\) outcomes. The pairs that sum to 7 are: \((1,6), (2,5), (3,4), (4,3), (5,2), (6,1)\) – six outcomes. \(\mathbb{P}(S = 7) = \dfrac{6}{36} = \dfrac{1}{6} \approx 16.67\%\). At least

Home › Math & Conversions › Core Math & Algebra › Dot Product Calculator Dot Product Calculator This dot product calculator works with 2D, 3D, and general n-dimensional vectors. Enter the components of two vectors and get the scalar product a·b , the length of each vector , the angle between them , and the projection of A onto B , with step-by-step expansion. It is designed for students and professionals in linear algebra, physics, engineering, and data science who need a precise and readable breakdown of vector computations. 1. Enter vectors Use comma or space separated values. The two vectors must have the same dimension. Examples: 1, 2, 3 , 0.5 -1.2 4.0 , 2 0 -3 5 . Vector A components Vector B components Decimal places Example 2D (orthogonal) Example 3D Clear inputs Calculate Clear results Supports any dimension with exact component product 2. Results Vector summary Dimension Dot product a·b |A| |B| cos(θ) Angle θ Orthogonality test Parallel

Home › Math & Conversions › Core Math & Algebra › Ellipse area Ellipse Area Calculator Compute the area of an ellipse using \(A = \pi a b\), from semi-axes or full diameters, with unit conversions, eccentricity, and equivalent circle radius. Core Math & Algebra Interactive ellipse area tool Select how your ellipse is specified, choose length units, and the calculator will evaluate the area \(A = \pi a b\) and key derived quantities. The tool handles both semi-axes and full diameters and reports results in several area units. Semi-axes a and b Full diameters (2a and 2b) Use semi-axes for direct \(A = \pi a b\), or diameters if you measured full width and height. Semi-major axis \(a\) Longest half-axis from center to edge. Semi-minor axis \(b\) Shortest half-axis from center to edge. Length units meters (m) centimeters (cm) millimeters (mm) kilometers (km) inches (in) feet (ft) yards (yd) Both axes must use the same unit. Output preferences Show bounding rectangle area (full width × full height) Show equivalent circle radius with the same area Calculate ellipse area Clear Example: \(a = 4\), \(b = 2\) Example: width 10 m, height 6 m Example: circle radius 3 m Results Input geometry (normalized) Area in primary units Area in SI & imperial units Shape metrics Bounding rectangle comparison Equivalent circle Formula for the area of an ellipse An ellipse is defined by two characteristic lengths: the semi-major axis \(a\) (the longest half-axis) and the semi-minor axis \(b\) (the shortest half-axis). Its area is given by the simple but powerful formula \(A = \pi a b.\) If you instead know the full diameters (also called the full major and minor axes), usually denoted by \(W = 2a\) and \(H = 2b\), first convert to semi-axes: \(a = \dfrac{W}{2}, \quad b = \dfrac{H}{2}, \quad A = \pi a b = \dfrac{\pi}{4} W H.\) Ellipse vs. circle – a special case A circle is just a special ellipse where the two semi-axes are equal: \(a = b = r\). In that case, the general formula reduces to the familiar circle area formula: \(A = \pi a b = \pi r^2.\) The calculator automatically detects when \(a\) and \(b\) are equal (within a small tolerance) and reports that you effectively have a circle. Eccentricity and how “stretched” the ellipse is Besides area, engineers and mathematicians often describe an ellipse using its eccentricity \(e\), a measure of how elongated it is. For an ellipse with \(a \ge b > 0\), the eccentricity is defined by \(e = \sqrt{1 - \dfrac{b^2}{a^2}}.\) Values of \(e\) range from \(e = 0\) for a perfect circle (\(a = b\)), to values very close to 1 for a highly stretched ellipse. Although eccentricity does not affect the area formula , it is useful for understanding the shape and for applications like orbital mechanics and beam cross-sections. Bounding rectangle comparison Any ellipse with semi-axes \(a\) and \(b\) fits exactly inside a bounding rectangle of width \(2a\) and height \(2b\). The area of this rectangle is \(A_{\text{rect}} = (2a)(2b) = 4ab.\) Comparing the ellipse area to its bounding rectangle: \(\dfrac{A_{\text{ellipse}}}{A_{\text{rect}}} = \dfrac{\pi ab}{4ab} = \dfrac{\pi}{4} \approx 0.785.\) So an ellipse covers about 78.5% of the area of its bounding rectangle, regardless of the specific values of \(a\) and \(b\). Equivalent circle radius Sometimes you may want to replace an ellipse by a circle with the same area. Setting the circle area \(\pi r^2\) equal to the ellipse area \(\pi a b\) gives: \(\pi r^2 = \pi a b \quad \Rightarrow \quad r = \sqrt{ab}.\) The calculator reports this equivalent radius , which is helpful when comparing elliptical ducts or tanks to circular ones with the same cross-sectional area. Ellipse area – FAQ Which values should I use for a and b? Always use the semi-axes : distances from the center to the ellipse in each principal direction. If you start from full diameters (end-to-end measurements), divide them by 2 to obtain a and b before applying the formula. The calculator can do this conversion automatically when you choose the diameter mode. Do the units of a and b matter? They matter only in the sense that both must be the same. If a and b are in meters, area will be in square meters; if in centimeters, area will be in square centimeters, and so on. This tool lets you select the length unit once and then shows the area in a range of common square units (m², cm², mm², ft²). Why is the area formula so simple compared with the perimeter? The area formula \(A = \pi a b\) has a clean closed form because it comes from integrating a quadratic curve. By contrast, the perimeter (circumference) of an ellipse does not have a simple elementary expression and is usually approximated using series or special functions. This calculator focuses on area, where an exact formula is available. Where is ellipse area used in practice? Ellipse areas appear in structural and mechanical engineering (beam and shaft cross-sections), ventilation and piping (ducts and conduits), architecture (arched openings, domes), and physics (phase-space plots, orbital cross-sections). Accurately computing the area helps with sizing, flow capacity, and stress calculations. Can I use this calculator for exam and design work? Yes, the tool is designed to match standard textbook formulas and unit practices. It can confirm hand calculations and provide quick checks during design. For formal design submissions, you should still document your assumptions and formula derivations, but this calculator gives a reliable numeric reference. Frequently Asked Questions How accurate is this ellipse area calculator? The area formula \(A = \pi a b\) is exact. The only rounding comes from displaying results to a limited number of decimal places and from floating-point arithmetic in your browser, which is more than sufficient for typical engineering, physics, and educational applications. Can I mix units for a and b? No. To avoid hidden conversion errors, the calculator assumes that both axes are expressed in the same length unit. Convert all measurements to a common unit (for example, meters) before entering them, or change the global unit selector and re-enter the values. Does this calculator handle tilted ellipses? Yes, as long as you know the lengths of the principal axes, any orientation in the plane gives the same area; rotation does not change area. The tool assumes you are supplying the semi-major and semi-minor axis lengths, not arbitrary chord lengths. Related Core Math & Algebra tools Explore more geometry and analysis tools that pair naturally with ellipse area. Pythagorean theorem Convolution Dice roll probability Continued fraction Exponent calculator Fibonacci number Sine, cosine, tangent (SOHCAHTOA) Laplace transform Prime number tools All Math & Conversions tools Quick checklist for ellipse area calculations Decide whether you are working with semi-axes or full diameters. Convert all length measurements to a consistent unit. Ensure you have correctly identified the major and minor axes. Apply \(A = \pi a b\) for semi-axes, or \(A = \dfrac{\pi}{4} W H\) for diameters. Compare the result with the bounding rectangle or an equivalent circle if helpful.

Home › Math & Conversions › Core Math & Algebra › Equation Solver Equation Solver Solve linear and quadratic equations, or small systems of equations, with step-by-step explanations and a numeric root finder for general equations of the form f(x) = 0. Single equation System of equations Equation in one variable (x) Use x as the variable and ^ for powers (e.g., x^2 ). You can also enter equations like sin(x) - x/2 = 0 for the numeric solver. Solving method Auto detect (linear/quadratic first) Linear (ax + b = c) Quadratic (ax^2 + bx + c = 0) Numeric (Newton–Raphson) Initial guess (numeric mode) Used only for the numeric solver. Tolerance Smaller values require more iterations. Solve equation Clear The solution and step-by-step explanation will appear here. Solve a small linear system in the form Ax = b. Enter the coefficients and constants for each equation. System size 2 × 2 system (variables x, y) 3 × 3 system (variables x, y, z) Equation x y z = Constant 1 = 2 = 3 = For a 2×2 system, only the first two rows and the x, y columns are used. The third row and z column are ignored. Solve system Clear The solution and intermediate matrix steps will appear here. How this equation solver works This tool is designed for core algebra and pre-calculus work. It combines exact algebraic formulas for linear and quadratic equations with a numeric solver for more general problems, plus a compact linear system solver for small systems. 1. Linear equations in one variable A linear equation in one variable has the form ax + b = c , where a , b , and c are real numbers and a ≠ 0 . Solving a linear equation: Start from: ax + b = c Subtract b from both sides: ax = c - b Divide by a : x = (c - b)

Home › Math & Conversions › Core Math & Algebra › Euclidean algorithm Euclidean Algorithm Calculator Compute the greatest common divisor (GCD), view the division steps, and run the extended Euclidean algorithm with Bézout coefficients and modular inverses. Core Math & Algebra Interactive Euclidean algorithm tool Enter two integers and choose between the basic GCD computation and the extended Euclidean algorithm. The tool shows each division step and, in extended mode, the coefficients \(x\) and \(y\) such that \(\gcd(a,b) = a x + b y\). You can also optionally compute a modular inverse. Basic GCD & division steps Extended algorithm & modular inverse First integer \(a\) Nonzero integer (positive or negative). Second integer \(b\) Nonzero integer (positive or negative). Modulus \(m\) for inverse (optional) Used only in extended mode; requires \(\gcd(a, m) = 1\). Run algorithm Clear Load example: \(\gcd(252, 198)\) GCD result & division table – Input pair Greatest common divisor Number of divisions Division steps (Euclidean algorithm) Step Dividend Divisor Quotient Remainder Equation Interpretation Extended Euclidean algorithm – Input pair GCD Bézout identity Coefficient evolution The table shows how coefficients \((x_i, y_i)\) evolve so that each remainder \(r_i = a x_i + b y_i\). The last nonzero remainder is the GCD with its final coefficients. Step Quotient Remainder x y Modular inverse Interpretation What is the Euclidean algorithm? The Euclidean algorithm is a classical method to compute the greatest common divisor (GCD) of two integers \(a\) and \(b\). It is based on the key observation that \[ \gcd(a, b) = \gcd(b, r) \] whenever \(a = qb + r\) with \(0 \le r < |b|\). In other words, replacing the larger number by the remainder of the division does not change the greatest common divisor. Repeating this process eventually produces a remainder of 0; the last nonzero remainder is the GCD. Algorithm (division-based GCD) Starting from two integers \(a\) and \(b\), not both zero: Set \(a_0 = |a|\), \(b_0 = |b|\). While \(b_k \ne 0\), compute the quotient and remainder: \[ a_k = q_k b_k + r_k,\quad 0 \le r_k < b_k. \] Set \(a_{k+1} = b_k\), \(b_{k+1} = r_k\). When \(b_k = 0\), the GCD is \(a_k\). The table in the calculator shows the sequence of quotients \(q_k\) and remainders \(r_k\) until the algorithm terminates. The extended Euclidean algorithm The extended Euclidean algorithm doesn’t just give \(g = \gcd(a,b)\); it also finds integers \(x\) and \(y\) such that \[ g = \gcd(a,b) = a x + b y. \] These are called Bézout coefficients . They are crucial when you need a modular inverse: if \(\gcd(a, m) = 1\), then the coefficient \(x\) in \(1 = a x + m y\) gives a modular inverse of \(a \bmod m\) (specifically, \(a^{-1} \equiv x \pmod m\)). Applications of the Euclidean algorithm Reducing fractions to lowest terms (e.g., \( \frac{a}{b} = \frac{a/g}{b/g} \) where \(g = \gcd(a,b)\)). Solving linear Diophantine equations \(a x + b y = c\). Computing modular inverses in cryptographic protocols. Understanding the structure of integer lattices and continued fractions. Euclidean algorithm – FAQ Does the Euclidean algorithm always terminate? Yes. At each step, the remainder is strictly smaller than the divisor, so the sequence of nonnegative remainders decreases until it reaches 0. This guarantees termination in a finite number of steps. Why is the last nonzero remainder the GCD? Each remainder is a linear combination of the original integers \(a\) and \(b\), so it is divisible by any common divisor of \(a\) and \(b\). On the other hand, the last nonzero remainder divides the previous remainder and thus divides all preceding remainders and the original inputs. Together, these facts show that this last remainder is the greatest common divisor. What happens if one of the inputs is 0? By convention, \(\gcd(a, 0) = |a|\) and \(\gcd(0, b) = |b|\) for nonzero \(a\) and \(b\). The calculator enforces that at least one of the inputs is nonzero; if one is 0, the algorithm effectively stops after one step. Is the Euclidean algorithm only for integers? The classical algorithm is formulated for integers. Variants exist for polynomials over fields and for other Euclidean domains, but this calculator focuses on integers, which are the standard setting in elementary number theory and cryptography. How large can the numbers be? This implementation uses JavaScript’s safe integer range (roughly up to \(9 \times 10^{15}\) in absolute value). For very large integers used in industrial cryptographic libraries, you would typically rely on big-integer implementations of the Euclidean algorithm. Frequently Asked Questions What is the difference between the basic and extended Euclidean algorithm? The basic algorithm only tracks remainders and computes the greatest common divisor. The extended algorithm additionally tracks coefficients for the inputs a and b so that each remainder can be written as a linear combination of a and b. The last nonzero remainder provides the GCD and its Bézout coefficients. How do I use the tool to find a modular inverse? Switch to the extended mode, enter a as the integer whose inverse you want, and m as the modulus. If gcd(a, m) = 1, the calculator will report an x such that a · x ≡ 1 (mod m). If the GCD is not 1, no modular inverse exists with respect to m. Why does the calculator use absolute values internally? The GCD is always taken to be nonnegative, and gcd(a, b) = gcd(|a|, |b|). Using absolute values simplifies the division steps without changing the result, while signs are taken into account when reporting the Bézout coefficients. Related Core Math & Algebra tools Complement the Euclidean algorithm with these number theory and algebra calculators. Set theory Interval notation Base converter Exponent Fibonacci number Prime number Continued fraction Dice roll probability Pythagorean theorem SOHCAHTOA (trigonometry) Convolution All Math & Conversions tools Quick checklist Ensure at least one of \(a\), \(b\) is nonzero. Optionally simplify by dividing both numbers by small common factors. Run the Euclidean algorithm and confirm the last nonzero remainder. Use extended mode when you need Bézout coefficients or modular inverses. Document steps if you are presenting the computation in proofs or exams.

Home › Math & Conversions › Core Math & Algebra › Exponent Exponent Calculator Compute \(a^b\) quickly and safely. Supports positive, zero, negative and fractional exponents, and shows results in standard and scientific notation with clear explanations. This page covers the mathematical meaning of exponents and exponentiation, not the interview platform or consulting firms that share the name “Exponent”. Core Math & Algebra Interactive exponentiation tool Enter a base \(a\) and an exponent \(b\) and the calculator will evaluate \(a^b\), check domain issues (like \(0^0\) or negative bases with non-integer exponents), and present the result in multiple formats. Base \(a\) Any real number. For negative bases, only integer exponents are allowed. Exponent \(b\) Accepts integers, decimals, and simple fractions like 1/2 or 3/4. Decimal places 4 decimals 8 decimals 12 decimals Only affects how the result is displayed, not the internal precision. Options Show repeated-multiplication steps for small integer exponents Always show scientific notation \(m \times 10^k\) Calculate \(a^b\) Clear Example: \(2^{10}\) Example: \(10^{-3}\) Example: \(9^{1/2}\) Results Main value Scientific notation Interpretation Step-by-step multiplication For larger exponents, repeated multiplication is not shown explicitly; the calculator still uses an efficient power algorithm internally. What is an exponent? In mathematics, an exponent tells you how many times to multiply a base by itself. The expression \(a^b\) is read “\(a\) to the power of \(b\)” or “\(a\) raised to \(b\)”. For example: \(2^3 = 2 \cdot 2 \cdot 2 = 8\) \(10^4 = 10\,000\) \(5^1 = 5\) Exponentiation extends this idea beyond simple repeated multiplication, allowing zero , negative , and even fractional exponents. Special exponent cases Zero and one \(a^1 = a\) for any real \(a\). \(a^0 = 1\) for any nonzero \(a\). This is consistent with the laws of exponents. \(0^0\) is indeterminate in many contexts and is left undefined in this calculator. Negative exponents A negative exponent means reciprocal . For a nonzero base \(a\) and positive integer \(n\): \(a^{-n} = \dfrac{1}{a^n}\). Examples: \(2^{-3} = 1/2^3 = 1/8 = 0.125\) \(10^{-2} = 1/100 = 0.01\) Fractional exponents Fractional exponents connect exponentiation with roots. For a positive base \(a\) and integers \(p\) and \(q>0\): \(a^{p/q} = \left(\sqrt[q]{a}\right)^p = \sqrt[q]{a^p}\). Examples: \(9^{1/2} = \sqrt{9} = 3\) \(27^{2/3} = (\sqrt[3]{27})^2 = 3^2 = 9\) \(16^{3/4} = (\sqrt[4]{16})^3 = 2^3 = 8\) Basic laws of exponents These rules hold for a wide range of exponents (with the usual domain caveats such as nonzero base when dividing): \(a^m \cdot a^n = a^{m+n}\)   (same base, add exponents) \(\dfrac{a^m}{a^n} = a^{m-n}\)   (subtract exponents) \((a^m)^n = a^{mn}\)   (power of a power) \((ab)^n = a^n b^n\)   (power of a product) \(\left(\dfrac{a}{b}\right)^n = \dfrac{a^n}{b^n}\)   (power of a quotient) The calculator respects these relationships when simplifying and interpreting your input. Exponents and scientific notation Exponents are central to scientific notation , which expresses numbers as \(m \times 10^k\) with \(1 \le |m| < 10\) and integer \(k\). This makes it easy to work with very large or very small values. For example: \(3\,000 = 3 \times 10^3\) \(0.000045 = 4.5 \times 10^{-5}\) The exponent calculator automatically converts your result into this form so you can quickly see the order of magnitude. Exponent – FAQ Are all combinations of base and exponent valid over the real numbers? No. For example, \(0^0\) and \(0^{-1}\) are undefined, and negative bases with non-integer exponents typically produce complex numbers. This tool focuses on real-valued exponentiation and will warn you when an input falls outside that domain. Why is 0 raised to a positive exponent allowed? For any positive exponent \(b\), \(0^b = 0\) is consistent with the limit behavior and algebraic rules. Problems appear only with zero or negative exponents on a zero base, where division by zero or indeterminate forms arise. How does this relate to logarithms? Logarithms are the inverse of exponentiation. If \(a^b = x\) with \(a>0\) and \(a \ne 1\), then \(b = \log_a x\). While this page focuses on computing powers, the same concepts underpin log and exponential functions used in finance, statistics, and physics. Does the calculator use exact arithmetic? Internally, it uses standard floating-point arithmetic, which is approximate. For many everyday values and exponents this is more than sufficient. The decimal-places setting only controls how the result is displayed, not the underlying precision. Can I use this for teaching and exam work? Yes. The tool is designed as a teaching aid: it highlights domain issues, shows scientific notation, and can display repeated-multiplication steps for small integer exponents. For formal work, you may still want to show the algebra manually, but the calculator is excellent for checking results. Frequently Asked Questions How accurate is this exponent calculator? It uses the browser’s IEEE-754 floating-point arithmetic, which is the same standard used in most scientific software and programming languages. Results are usually accurate to many decimal places; any rounding errors are far smaller than typical measurement or modelling uncertainties. What should I watch out for when using very large exponents? Extremely large positive exponents can overflow to infinity, while very negative exponents can underflow to zero. When that happens, the calculator will warn you. In such regimes, using logarithms or symbolic manipulation is often more appropriate. Can this calculator handle complex exponents? No, this version is restricted to real-valued exponentiation. Complex exponents require careful branch choices and complex analysis results, which go beyond the scope of a general-purpose calculator aimed at everyday math, science, and engineering users. Related Core Math & Algebra tools Explore more algebra and function tools that naturally pair with exponents and powers. Power & root calculator Logarithm calculator Scientific notation Exponential growth rate Modular arithmetic Set theory Interval notation Continued fraction Convolution All Math & Conversions tools Checklist for safe exponent use Check whether the base is zero, negative, or positive. Identify if the exponent is integer, negative, or fractional. For negative bases, ensure the exponent is an integer. Avoid undefined forms like \(0^0\) or \(0^{-1}\). Use scientific notation when communicating very large or small results.

Home › Math & Conversions › Core Math & Algebra › Fibonacci number Fibonacci Number Calculator Compute the n-th Fibonacci number exactly, list the sequence up to n, and see how fast the ratio F n+1 /F n approaches the golden ratio. Core Math & Algebra Interactive Fibonacci number tool Enter an index \(n\) and this tool will compute the Fibonacci number \(F_n\) using an efficient fast-doubling algorithm with exact big-integer arithmetic. You can also display the sequence from \(F_0\) up to \(F_n\) and, for moderate \(n\), the ratio \(F_{n+1}

Home › Math & Conversions › Core Math & Algebra › FOIL Method Calculator FOIL Method Calculator – Expand & Factor This FOIL method calculator helps you expand products of binomials \((ax + b)(cx + d)\) and factor quadratics \(ax^2 + bx + c\) using the reverse FOIL approach. It shows every step – First, Outer, Inner, Last – and gives the simplified polynomial, so you can check homework, build intuition, or prepare lesson material. FOIL method calculator Mode Expand (ax + b)(cx + d) Factor ax² + bx + c (reverse FOIL) Variable symbol Usually x, y, t… Single letter recommended. Enter the coefficients of your binomials in the form (a·x + b)(c·x + d) . Coefficients can be negative or decimal. First binomial (ax + b) a (coefficient of variable) b (constant term) Second binomial (cx + d) c (coefficient of variable) d (constant term) You can also type non-integer values (e.g. 1.5, -0.25). Use dot or comma as decimal separator. Variable symbol Usually x, y, t… Single letter recommended. Enter coefficients of your quadratic \(ax^2 + bx + c\). The tool searches for an integer factorization of the form \((p x + q)(r x + s)\). a (coefficient of x²) b (coefficient of x) c (constant term) If no integer factorization is found, the calculator tells you and suggests using the quadratic formula. Calculate Clear Output is shown as both text and MathJax-formatted formulas. Result Step-by-step What is the FOIL method? The FOIL method is a mnemonic for multiplying two binomials. FOIL stands for: F irst – multiply the first terms in each binomial O uter – multiply the outer pair of terms I nner – multiply the inner pair of terms L ast – multiply the last terms in each binomial For binomials \((ax + b)\) and \((cx + d)\), FOIL gives: \( (ax + b)(cx + d) = \underbrace{acx^2}_{\text{First}} + \underbrace{adx}_{\text{Outer}} + \underbrace{bcx}_{\text{Inner}} + \underbrace{bd}_{\text{Last}} \) So the simplified form is \( acx^2 + (ad + bc)x + bd \). FOIL is just the distributive property Conceptually, FOIL is nothing more than applying the distributive property twice: \( (ax + b)(cx + d) = ax(cx + d) + b(cx + d) \) \( = ax \cdot cx + ax \cdot d + b \cdot cx + b \cdot d \). FOIL is handy for binomials, but for larger expressions (like trinomials) it is clearer to think in terms of full distribution. Reverse FOIL: factoring quadratics Factoring a quadratic polynomial \( ax^2 + bx + c \) into the product of two binomials \((px + q)(rx + s)\) is sometimes called reverse FOIL : \( (px + q)(rx + s) = prx^2 + (ps + qr)x + qs \) \( pr = a \) \( ps + qr = b \) \( qs = c \) To factor \( ax^2 + bx + c \) using integer reverse FOIL, you look for integers \( p, q, r, s \) that satisfy those three relationships. The calculator automates that search and either returns a neat factorization or tells you that no integer factorization exists. FOIL method – worked example Expand \((2x + 3)(x - 5)\). First: \( 2x \cdot x = 2x^2 \) Outer: \( 2x \cdot (-5) = -10x \) Inner: \( 3 \cdot x = 3x \) Last: \( 3 \cdot (-5) = -15 \) Now add all terms and combine like terms: \( (2x + 3)(x - 5) = 2x^2 - 10x + 3x - 15 = 2x^2 - 7x - 15 \). FOIL method – frequently asked questions Can I use FOIL for (x + 1)(x² + 2x + 3)? + No – FOIL is specifically for multiplying two binomials . For \((x + 1)(x^2 + 2x + 3)\) you should use the distributive property: \( (x + 1)(x^2 + 2x + 3) = x(x^2 + 2x + 3) + 1(x^2 + 2x + 3) \), then expand each part and combine like terms. The idea is the same, but FOIL as a mnemonic doesn’t cover trinomials directly. What if some coefficients are fractions? + FOIL works with any real numbers, including fractions and decimals. Just multiply normally and simplify at the end. This calculator accepts decimal input (for example 0.5 or -1.25) and returns a simplified decimal result. Does the order of the binomials matter? + No. Multiplication is commutative, so \((ax + b)(cx + d)\) and \((cx + d)(ax + b)\) expand to the same result. You may get the terms in a different intermediate order, but the simplified polynomial is identical. How is reverse FOIL related to the quadratic formula? + Reverse FOIL is a structured guess-and-check method to factor quadratics with “nice” integer roots. The quadratic formula works for all quadratics, and it will find roots even when no integer factorization exists. If reverse FOIL fails but you still need the exact roots, use the quadratic formula instead. Core Math & Algebra tools Basic Arithmetic Standard Form Calculator Percent Error Calculator Effect Size (Cohen’s d) G-test (Likelihood-Ratio Test) FOIL Method Calculator – you are here Graph Adjacency Matrix Speed, Distance, Time Friedman Test More algebra & polynomial tools Quadratic Equation Solver Polynomial Long Division Polynomial Factorization Linear Equation Solver System of Equations Solver Exponent & Radical Calculator

Home › Math & Conversions › Core Math & Algebra › Friedman test Friedman Test Calculator Run the Friedman test for repeated-measures designs. Enter either raw data (one row per subject/block, one column per treatment) or rank sums , and get the Friedman chi-square statistic, p-value and Kendall’s W effect size. Suitable for nonparametric analysis of k ≥ 3 related conditions when assumptions for repeated-measures ANOVA are doubtful. Input mode Raw data (recommended) Rank sums Blocks

Home › Math & Conversions › Core Math & Algebra › G-test (likelihood-ratio test) G-test Calculator (Likelihood-Ratio Test) This calculator implements the statistical G-test (likelihood-ratio test) for categorical data. It is not related to medical or prenatal screening products sometimes called “G-test”. Use it to compute the G statistic, degrees of freedom and p-value for goodness-of-fit problems and contingency tables (tests of independence). G-test calculator Test type Goodness-of-fit Independence (contingency table) All inputs are counts (frequencies). Use comma or dot as decimal separator; counts are rounded to non-negative numbers. Number of categories Between 2 and 10 categories. Expected pattern under H₀ Equal probabilities (uniform) Custom expected counts For “custom expected”, the calculator rescales your expected counts to match the total sample size. Observed counts Expected counts (before rescaling) Parameters estimated from data df = k − 1 − parameters. For many textbook problems this is 0. Context

Home › Math & Conversions › Core Math & Algebra › Grade Grade Calculator Compute your course grade from points or weighted categories, and see the corresponding letter grade and GPA-style value. Core Math & Algebra Interactive course grade calculator Choose between a points-based model (typical when your syllabus lists point values for each assignment) and a weighted model (when categories like exams or homework have a fixed percentage weight). Points-based (earned

Home › Math & Conversions › Core Math & Algebra › Graph adjacency matrix Adjacency Matrix Calculator for Graphs This interactive tool helps you build and analyze the adjacency matrix of a graph. Choose the number of vertices, toggle directed vs undirected and weighted vs unweighted , edit the matrix or paste an edge list, and get degrees, edge counts, density and adjacency lists. It is designed for students, data scientists and engineers who want a transparent, step-by-step companion to standard graph theory references. Adjacency matrix builder Number of vertices (n) Between 2 and 12 vertices. Graph type Undirected Directed Undirected graphs have symmetric adjacency matrices when unweighted. Weights Unweighted (0/1) Weighted In weighted mode, entries represent edge weights; 0 means no edge. Vertex labels (optional, comma-separated) If empty, vertices are labeled v1, v2, …, vn. Labels are used in adjacency lists and edge parsing. Build

Home › Math & Conversions › Core Math & Algebra › Hyperbolic Function Explorer Hyperbolic Function Calculator & Explorer Evaluate hyperbolic functions like sinh, cosh, tanh and their inverses, see exponential definitions, and explore key identities with a quick reference table. All functions are evaluated for a real variable x . Hyperbolic functions take a dimensionless argument. Hyperbolic function calculator Choose a function, enter a value for x , and click Calculate . The tool evaluates the function numerically and compares equivalent definitions. Function sinh(x) = (e^x − e^{−x})

Home › Math & Conversions › Core Math & Algebra › Interval Notation Interval Notation Calculator Convert inequalities, endpoints, and interval notation in one place. See the result on a number line and get the corresponding set-builder notation explained step by step. Inequality → Interval Interval → Inequality Endpoints → Interval Enter an inequality in x Supported patterns include: x > a , x ≥ a , x < b , x ≤ b , and double inequalities like a < x ≤ b . Convert to interval notation Enter an interval This tool parses a single interval such as (a, b) , [a, b] , (−∞, b] , or [a, ∞) . Unions like (-∞, 0) ∪ (1, ∞) are not yet supported. Convert to inequalities Left endpoint ( a ) Right endpoint ( b ) Left side ( open (not included) [ closed (included) Right side ) open (not included) ] closed (included) Leave an endpoint blank if it is not bounded on that side and choose the appropriate parenthesis for (−∞ or ∞) . Build interval Result Interval notation — Inequality form — Set-builder notation — Plain-language description Enter an inequality, interval, or endpoints above to see a detailed explanation here. Number line visualization The shaded segment shows the interval on a simplified real number line. Open circles represent excluded endpoints; filled circles represent included endpoints. What is interval notation? Interval notation is a compact way to describe sets of real numbers. Instead of writing “all real numbers x such that -1 < x ≤ 3 ”, we can write the interval (-1, 3] . Basic types of intervals (a, b) : open interval, excludes both endpoints a and b . [a, b] : closed interval, includes both endpoints. [a, b) or (a, b] : half-open (half-closed) intervals. (−∞, b) , (−∞, b] : intervals unbounded on the left. (a, ∞) , [a, ∞) : intervals unbounded on the right. How to convert inequalities to interval notation 1. Single-ended inequalities Consider inequalities involving a single bound: x > a → (a, ∞) x ≥ a → [a, ∞) x < b → (−∞, b) x ≤ b → (−∞, b] 2. Double inequalities For a double inequality like a < x ≤ b , the middle term tells you the variable ( x ), and the inequalities tell you about inclusion or exclusion: If the inequality is strict (< or >), use parentheses: ( ) . If the inequality is inclusive (≤ or ≥), use brackets: [ ] . Example: -1 < x ≤ 3 becomes the interval (-1, 3] . From interval notation back to inequalities To reverse the process, read the interval from left to right: (a, b) means a < x < b . [a, b] means a ≤ x ≤ b . (−∞, b] means x ≤ b . [a, ∞) means x ≥ a . Interval notation, sets, and domains In calculus and algebra, interval notation is often used to describe the domain and range of functions. For example, the function f(x) = 1/x has domain (-∞, 0) ∪ (0, ∞) : all real numbers except 0 . In set-builder notation, you might see { x ∈ ℝ | x > 2 } , which is equivalent to the interval (2, ∞) . Set-builder and interval notation { x ∈ ℝ | a < x < b } ↔ (a, b) { x ∈ ℝ | a ≤ x ≤ b } ↔ [a, b] { x ∈ ℝ | x ≥ a } ↔ [a, ∞) { x ∈ ℝ | x < b } ↔ (−∞, b) Interval notation FAQ What do the different brackets mean? Parentheses ( ) always mean “open” or “not included”, while brackets [ ] mean “closed” or “included”. Infinity symbols ∞ and −∞ are always paired with parentheses, because infinity itself is not a real number. Can an interval be empty? Yes. For example, the condition x < 1 and x ≥ 1 at the same time describes the empty set. In practice, inconsistent inequalities lead to no interval; this calculator reports such cases as invalid input. What about unions like (-∞, 0) ∪ (1, ∞)? Many real-world domains are a union of disjoint intervals. This calculator currently focuses on a single interval at a time. You can handle unions by converting each piece separately and then combining them with the union symbol ∪ in your work. Frequently Asked Questions How is interval notation different from inequalities? Interval notation compresses an entire inequality (or system of inequalities in one variable) into a short expression using brackets and parentheses. Both notations describe the same set; interval notation is just more compact and easier to scan. Do I always use ℝ (real numbers) in set-builder notation? For most introductory algebra and calculus problems, the underlying set is the real numbers ℝ. In more advanced settings you might restrict to integers ℤ or other sets, but interval notation itself is designed for real intervals. How do I write the domain of a function in interval notation? Start by identifying values that must be excluded (for example, denominators equal to zero or even roots of negative numbers). Then express the remaining real numbers as one or more intervals, using parentheses or brackets according to whether the endpoints are included. Core Math & Algebra tools Set theory Set operations Venn diagram generator Base converter Modular arithmetic Modulo Equation solver Number theory Graphing Probability Prime number Interval notation (current) Study tips for interval notation Always sketch a quick number line to visualize the set. Match strict inequalities with parentheses and inclusive ones with brackets. Remember that infinity symbols are never included in the set. Practice converting domains and ranges between graphs, inequalities, and intervals.

Home › Math & Conversions › Core Math & Algebra › Laplace transform Laplace Transform Calculator Compute Laplace transforms for standard time-domain functions, see the formulas and region of convergence (ROC), and get numeric values at a chosen real s 0 . Core Math & Algebra Interactive Laplace transform tool Choose a standard time-domain signal \(f(t)\), set its parameters, and let the calculator produce the symbolic Laplace transform \(F(s)\), with a short explanation and region of convergence. Optionally, evaluate \(F(s)\) at a specific real value \(s_0\). Standard Laplace pairs Custom ODE solver (coming soon) Time-domain function \(f(t)\) Exponential: \(f(t) = K e^{a t} u(t)\) Power: \(f(t) = K t^n u(t)\) Sine: \(f(t) = K \sin(b t) u(t)\) Cosine: \(f(t) = K \cos(b t) u(t)\) Damped sine: \(f(t) = K e^{a t} \sin(b t) u(t)\) Damped cosine: \(f(t) = K e^{a t} \cos(b t) u(t)\) Shifted: \(f(t) = K u(t-a_0) g(t-a_0)\) Impulse: \(f(t) = K \delta(t-a_0)\) All signals are assumed causal (zero for \(t < 0\)) unless explicitly shifted by \(a_0\). Optional numeric evaluation \(s_0\) (real) If provided and within the ROC, the tool evaluates \(F(s_0)\) numerically. Amplitude \(K\) Real gain (can be negative). Exponential rate \(a\) Real \(a\). For a stable decaying exponential, use \(a < 0\). Notes \(f(t) = K e^{a t} u(t)\) → \(F(s) = \dfrac{K}{s - a}\), ROC: \(\mathrm{Re}(s) > \mathrm{Re}(a)\). Amplitude \(K\) Real gain. Power \(n\) (integer \(\ge 0\)) Common cases: \(n = 0,1,2,\dots\). Notes \(f(t) = K t^n u(t)\) → \(F(s) = K \dfrac{n!}{s^{n+1}}\), ROC: \(\mathrm{Re}(s) > 0\). Amplitude \(K\) Angular frequency \(b > 0\) \(b\) in rad/s (or the same unit as \(t^{-1}\)). \(f(t) = K \sin(bt) u(t)\) → \(F(s) = K \dfrac{b}{s^2 + b^2}\), ROC: \(\mathrm{Re}(s) > 0\). Amplitude \(K\) Angular frequency \(b > 0\) \(f(t) = K \cos(bt) u(t)\) → \(F(s) = K \dfrac{s}{s^2 + b^2}\), ROC: \(\mathrm{Re}(s) > 0\). Amplitude \(K\) Exponential rate \(a\) Angular frequency \(b > 0\) \(K e^{at} \sin(bt) u(t) \rightarrow F(s) = K \dfrac{b}{(s-a)^2 + b^2}\), ROC: \(\mathrm{Re}(s) > \mathrm{Re}(a)\). Amplitude \(K\) Exponential rate \(a\) Angular frequency \(b > 0\) \(K e^{at} \cos(bt) u(t) \rightarrow F(s) = K \dfrac{s-a}{(s-a)^2 + b^2}\), ROC: \(\mathrm{Re}(s) > \mathrm{Re}(a)\). Amplitude \(K\) Shift \(a_0 \ge 0\) Start time of the shifted signal. \(G(s)\) (Laplace of \(g(t)\)) Symbolic placeholder for the base transform \(G(s)\). If \(f(t) = K u(t-a_0) g(t-a_0)\) and \(\mathcal{L}\{g(t)\} = G(s)\), then \(\mathcal{L}\{f(t)\} = K e^{-a_0 s} G(s)\) with ROC equal to that of \(G(s)\). Amplitude \(K\) Shift \(a_0 \ge 0\) \(\mathcal{L}\{K \delta(t-a_0)\} = K e^{-a_0 s}\), ROC: all complex \(s\). Compute Laplace transform Clear Load example: damped sine Result summary Time-domain signal \(f(t)\) Laplace transform \(F(s)\) Region of convergence (ROC) Numeric evaluation at \(s_0\) Definition of the Laplace transform The unilateral Laplace transform of a time-domain function \(f(t)\), assumed zero for \(t < 0\), is defined as \(\displaystyle \mathcal{L}\{f(t)\}(s) = F(s) = \int_{0^-}^{\infty} f(t)\,e^{-st}\,\mathrm{d}t\) Here \(s\) is a complex variable, \(s = \sigma + j\omega\). For some functions the integral converges only for a subset of the complex plane, called the region of convergence (ROC). In control, circuits, and mechanical systems, the Laplace transform is a standard tool to turn differential equations into simple algebra in the \(s\)-domain, where transfer functions and poles/zeros are easier to interpret. Standard Laplace transform pairs The table below collects a few core Laplace transform pairs used most often in introductory courses, signal processing, and control engineering. \(f(t)\) \(F(s) = \mathcal{L}\{f(t)\}\) ROC \(u(t)\) \(\dfrac{1}{s}\) \(\mathrm{Re}(s) > 0\) \(e^{at} u(t)\) \(\dfrac{1}{s - a}\) \(\mathrm{Re}(s) > \mathrm{Re}(a)\) \(t^n u(t)\), \(n \in \mathbb{N}_0\) \(\dfrac{n!}{s^{n+1}}\) \(\mathrm{Re}(s) > 0\) \(\sin(bt) u(t)\) \(\dfrac{b}{s^2 + b^2}\) \(\mathrm{Re}(s) > 0\) \(\cos(bt) u(t)\) \(\dfrac{s}{s^2 + b^2}\) \(\mathrm{Re}(s) > 0\) \(e^{at}\sin(bt) u(t)\) \(\dfrac{b}{(s-a)^2 + b^2}\) \(\mathrm{Re}(s) > \mathrm{Re}(a)\) \(e^{at}\cos(bt) u(t)\) \(\dfrac{s-a}{(s-a)^2 + b^2}\) \(\mathrm{Re}(s) > \mathrm{Re}(a)\) \(u(t-a)f(t-a)\) \(e^{-as} F(s)\) Same ROC as \(F(s)\) \(\delta(t-a)\) \(e^{-as}\) All \(s\) ROC and stability intuition The region of convergence encodes how quickly the signal decays (or grows) and determines whether the Laplace transform exists. For example, a causal exponential \(e^{at}u(t)\) only has a finite transform if \(e^{(a-\sigma)t}\) decays as \(t \rightarrow \infty\), which requires \(\mathrm{Re}(s) = \sigma > \mathrm{Re}(a)\). In systems theory, the ROC relates directly to stability : if all poles of a causal LTI system lie in the left half-plane (\(\mathrm{Re}(s) < 0\)), then the ROC includes the imaginary axis and the system is BIBO stable. Laplace transform – FAQ Why is the Laplace transform so useful for differential equations? The Laplace transform converts differentiation into multiplication by \(s\): \(\mathcal{L}\{f'(t)\} = sF(s) - f(0^-)\). Higher-order derivatives lead to higher powers of \(s\). This turns linear constant-coefficient differential equations into algebraic equations in \(s\), which are much easier to solve; you then apply the inverse transform to return to \(f(t)\). What is the difference between unilateral and bilateral Laplace transforms? The unilateral (one-sided) Laplace transform integrates from \(0^-\) to \(\infty\) and is tailored to causal signals and initial-value problems. The bilateral (two-sided) transform integrates from \(-\infty\) to \(\infty\) and is useful when signals have non-zero values for negative time. This calculator follows the common unilateral convention. Can I combine results for sums of signals? Yes. The Laplace transform is linear: \(\mathcal{L}\{a f(t) + b g(t)\} = a F(s) + b G(s)\). You can decompose a complicated \(f(t)\) into a sum of standard building blocks, apply the table or the calculator to each term, then add the corresponding \(F(s)\) expressions. How accurate are the numeric evaluations? The numeric values use standard double-precision floating point, comparable to scientific calculators and engineering software. For extreme parameter values (very large or small magnitudes) rounding and overflow may occur; these limitations are inherent to finite-precision arithmetic rather than the transform formulas themselves. Frequently Asked Questions Do I always need the Laplace transform, or is the Fourier transform enough? The Fourier transform assumes signals are sufficiently well-behaved over all time and often focuses on steady-state sinusoidal behaviour. The Laplace transform generalizes Fourier by including exponential growth/decay and transient behaviour through the real part of s. In control and transient analysis, the Laplace transform is usually the more natural tool. Does this calculator compute inverse Laplace transforms? This version focuses on forward Laplace transforms (from \(f(t)\) to \(F(s)\)) with clear ROC descriptions. For many engineering problems, forward transforms combined with partial fraction decompositions and standard tables are sufficient. A future update may include a guided inverse-transform helper for simple rational \(F(s)\). How should I document Laplace transform steps in reports or lab work? When you use transform methods in formal work, always show the original differential equation, the initial conditions, the expression for \(F(s)\), and the corresponding inverse transform step. This calculator can help you verify each transform pair numerically, but your report should still include the full reasoning and references to standard tables. Related Core Math & Algebra tools Explore more tools that support calculus, algebra, and probability work alongside Laplace transforms. Convolution visualizer Pythagorean theorem SOHCAHTOA (trig ratios) Exponent rules Continued fraction Fibonacci number Dice roll probability Base converter Set theory basics Interval notation All Math & Conversions tools Checklist: using Laplace transforms safely Clarify whether you are using the unilateral or bilateral definition. Ensure \(f(t)\) is well-defined for \(t \ge 0\) and note any discontinuities or impulses. Write down the standard transform pair you are using and check its ROC. For sums of terms, apply linearity and transform each term separately. When solving ODEs, always apply the inverse transform and verify initial conditions are satisfied.

Home › Math & Conversions › Core Math & Algebra › Linear Algebra Toolbox Linear Algebra Toolbox – Matrix Calculator & System Solver Enter a 2×2, 3×3, or 4×4 matrix to compute determinant, rank, trace, inverse (when it exists), and solve linear systems of the form Ax = b . All computations are numerical and performed locally in your browser using double-precision floating-point arithmetic. Matrix calculator & Ax = b solver Choose the matrix size, enter coefficients, and (optionally) a right-hand-side vector b . Then click Calculate to compute key linear algebra quantities. Matrix size (n × n) 2 × 2 3 × 3 4 × 4 Small matrices keep results readable and highlight core concepts. System vector b (optional) Leave all cells empty to skip solving Ax = b . Matrix A entries Empty cells are treated as 0. Use decimals with a dot (e.g., 1.5). Negative values are allowed. Calculate Clear inputs Numerical tolerance for rank/singularity: 1e-10 Determinant, rank, trace, inverse (if it exists), and system solution details will appear here after you click Calculate . What is linear algebra used for? Linear algebra is the mathematical framework for working with linear relationships. Vectors and matrices capture systems of equations, geometric transformations, and data in high dimensions. Behind many modern technologies—computer graphics, recommendation systems, machine learning, optimization, and control theory—there is a layer of linear algebra doing the heavy lifting. 1. Matrices and linear systems A system of linear equations in \(n\) variables can be written compactly as \[ A x = b, \] where \(A\) is an \(n \times n\) matrix of coefficients, \(x\) is the unknown vector, and \(b\) is the right-hand-side vector. Solving the system means finding all vectors \(x\) that satisfy the equation. If \(A\) is invertible, the solution is unique and can be written as \(x = A^{-1} b\). When \(A\) is singular or nearly singular, the solution may not exist or may not be unique. 2. Determinant, rank, and trace Determinant \(\det(A)\) measures how volumes scale under the linear transformation represented by \(A\). A zero determinant means the transformation collapses some dimension and \(A\) is singular (not invertible). Rank is the dimension of the image of \(A\); it tells you how many independent directions survive after applying the transformation. For an \(n \times n\) matrix, full rank means rank \(n\). Trace \(\operatorname{tr}(A)\) is the sum of diagonal elements of \(A\). For many matrices, it also equals the sum of eigenvalues (counted with multiplicity). 3. Inverse matrices and singularity A square matrix \(A\) is invertible if there exists another matrix \(A^{-1}\) such that \(A A^{-1} = A^{-1} A = I\), where \(I\) is the identity matrix. Numerically: If \(\det(A)\) is very close to zero, \(A\) is ill-conditioned: small changes in the data can cause large changes in the solution. If the rank of \(A\) is less than \(n\), then \(A\) is singular and has no inverse. The toolbox uses Gaussian elimination with partial pivoting to estimate rank, determinant, and, where stable, the inverse and system solution. 4. Eigenvalues and beyond Eigenvalues and eigenvectors describe directions that a linear map scales without rotating. They are fundamental in dynamical systems, principal component analysis (PCA), and spectral graph theory. For clarity and robustness, this introductory toolbox focuses on determinant, rank, trace, and solving small systems, which are the building blocks you will typically meet first in a linear algebra course. Linear algebra toolbox – FAQ How should I interpret the determinant being “very small” but not exactly zero? + If the determinant is tiny relative to the scale of the entries, the matrix is nearly singular . Numerically, this means solving Ax = b or computing A −1 will be sensitive to rounding and data errors. The toolbox flags such situations by using a small tolerance when deciding whether a pivot is effectively zero. What does it mean if the tool says that the system Ax = b is inconsistent? + An inconsistent system is one where no vector x satisfies all the equations simultaneously. In row-echelon form, this appears as a row of the form [0 0 … 0 | c] with c ≠ 0 in the augmented matrix [A | b]. In that case, the tool reports that the system has no solution under the given numeric tolerance. Why does the tool limit matrices to 4 × 4? + The focus of this page is clarity and reliability for hand-sized problems. For larger matrices, numerical stability, performance, and visualization require more sophisticated interfaces and back-end libraries. For study, exam preparation, or small engineering checks, 2×2–4×4 is typically enough to illustrate the main ideas. Core Math & Algebra tools Modular Arithmetic Equation Solver Circular Segment Area Trig Identity Explorer Hyperbolic Function Explorer Linear Algebra Toolbox (you are here) Trapezoidal Rule Perfect Number Checker Chinese Remainder Theorem Set Theory Basics Geometry & analysis tools Pythagorean Theorem Convolution Explorer Dice Roll Probability Base Converter Interval Notation Helper Professional use note This linear algebra toolbox is ideal for education, exam practice, and quick engineering checks. For large-scale or safety-critical applications, verify key results with independent numerical software or formal analysis.

Home › Math & Conversions › Core Math & Algebra › Lottery Odds Calculator Lottery Odds Calculator Compute exact lottery odds for standard number-drawing games. Define the game rules and this calculator will return the probability of matching 0, 1, 2, … up to all numbers, plus the familiar “1 in X” jackpot odds . The engine uses hypergeometric probability behind the scenes and can handle both single-drum lotteries (classic 6-from-49 style) and two-drum lotteries with bonus balls (Powerball-style games). 1. Choose lottery type Lottery structure Single-drum (one number pool) Two-drum (main pool + bonus ball) Single-drum: all numbers are drawn from the same pool. Two-drum: main numbers + separate bonus/Powerball pool. 2A. Single-drum lottery parameters Example: “Choose 6 numbers out of 49; draw 6 winning numbers out of the same 49”. Numbers in pool (N) Total distinct numbers in the game (e.g. 49). Numbers on your ticket (T) How many distinct numbers you choose. Numbers drawn (K) How many numbers the lottery draws. Min. matches for a win (r min ) Used for “P(matches ≥ r min )” in the summary. Decimals in output 2B. Two-drum lottery parameters Example: “Pick 5 main numbers from 1–69 and 1 bonus ball from 1–26; draw 5 main + 1 bonus”. Main number pool Numbers in main pool (N₁) Numbers on ticket (T₁) Main numbers drawn (K₁) Bonus

Home › Math & Conversions › Core Math & Algebra › Matrix Determinant Calculator Matrix Determinant Calculator This matrix determinant calculator computes det(A) for 2×2, 3×3 and general n×n matrices (up to 6×6). It uses Gaussian elimination with partial pivoting for robust results and shows a row-operation log so you can follow every step. Use it to check theoretical exercises, verify hand computations, or quickly test whether a matrix is invertible (non-singular) . Ideal for linear algebra, numerical methods, and applied mathematics courses. 1. Define the matrix Matrix size n×n 2 × 2 3 × 3 4 × 4 5 × 5 6 × 6 Decimal places Supports real-valued entries Matrix A Example 2×2 Example 3×3 Example 4×4 (sparse) Clear matrix Tip: use Tab

Home › Math & Conversions › Core Math & Algebra › Modulo Modulo Calculator Compute a mod n, return the integer remainder of division, handle negative numbers, and check modular congruence. Includes a batch mode for modulo tables. Core Math & Algebra Interactive modulo and remainder calculator Use this tool to compute the remainder of integer division (a mod n), understand Euclidean division a = q·n + r, and verify congruences a ≡ b (mod n). The calculator always uses the Euclidean definition of modulo, so the remainder r is between 0 and n − 1 for n > 0, even if a is negative. Single value Batch list (table) Dividend a Integer to be divided You can enter positive or negative integers. Modulus n Positive integer divisor Must be a positive integer. The calculator enforces n > 0. Optional comparison b Check a ≡ b (mod n) Leave empty if you only need a single modulo result. List of integers Values a i (comma, space, or line separated) The tool will ignore empty entries. All values must be integers. Modulus n Positive integer divisor Ideal for building modulo tables or checking repeating patterns. Compute modulo Clear Load example Modulo result (single value) Valid input Dividend a Modulus n Quotient q Remainder r (a mod n) By Euclidean division: Range This calculator always returns r with 0 ≤ r < n. Congruence check a ≡ b (mod n) Comparison b a mod n b mod n Relation Modulo table (batch mode) Table generated Each row shows a value a i , the integer quotient q i , and the remainder r i = a i mod n. a i n q i r i = a i mod n Identity What is modulo? In arithmetic, the modulo (or mod ) operation returns the remainder when one integer is divided by another positive integer. When we write \[ a \bmod n, \] we mean the remainder r that appears in the Euclidean division of a by n: \[ a = q n + r, \quad \text{with } q \in \mathbb{Z},\; 0 \le r < n. \] For example, 17 mod 5 = 2 because 17 = 3 × 5 + 2 and 2 lies in the interval [0, 5). The calculator implements exactly this definition. Euclidean division and remainder For integers a and n > 0, Euclid's division theorem guarantees that there exist unique integers q (quotient) and r (remainder) such that: \[ a = q n + r \quad \text{and} \quad 0 \le r < n. \] In this calculator: You enter a and n as integers with n > 0. The tool solves for q and r. The displayed value of a mod n is exactly r. If a is negative, q will also be negative in such a way that r stays between 0 and n − 1. For instance, −7 mod 5 = 3 because −7 = (−2) × 5 + 3. Modulo with negative numbers When you compute modulo by hand or in pure math, the remainder is defined to be non-negative and less than n. However, some programming languages use a raw remainder operator that follows the sign of the dividend, which can create confusion when a is negative. This calculator always follows the Euclidean convention: \[ r = a - n \left\lfloor \frac{a}{n} \right\rfloor, \quad 0 \le r < n. \] That is why you may see results like −1 mod 5 = 4 or −7 mod 5 = 3. These are the standard values in modular arithmetic and make many theorems and identities work cleanly. Modular congruence a ≡ b (mod n) Two integers a and b are said to be congruent modulo n if they leave the same remainder when divided by n. Formally: \[ a \equiv b \pmod{n} \quad \Longleftrightarrow \quad n \mid (a - b) \quad \Longleftrightarrow \quad a \bmod n = b \bmod n. \] Typical examples: 17 ≡ 2 (mod 5), because both have remainder 2 mod 5. −1 ≡ 4 (mod 5), because −1 mod 5 = 4. 8 ≡ 0 (mod 4), because 4 divides 8 exactly. The congruence checker in this calculator compares a and b modulo n and reports whether they are congruent, including their canonical remainders. Practical uses of modulo Modulo arithmetic appears in many applications: Clock arithmetic: hours wrap around every 12 or 24. Computer science: hashing, cyclic buffers, and random number generators. Number theory: studying primes, residues, and patterns in integers. Scheduling and repetition: events occurring every n days, positions in cycles. The batch table mode helps you spot repeating patterns such as cycles of residues in these contexts. Modulo calculator – FAQ Can I use non-integer numbers in the modulo calculator? No. The standard definition of modulo is based on integer division, so this calculator requires that a and n be integers and n > 0. If you need floating-point remainders, consider scaling your values to integers or using a different tool tailored to real numbers. What happens if I enter n = 0? Division by zero is undefined, so modulo with n = 0 is also undefined. The calculator will return an error and ask you to choose a positive integer modulus instead. Why do some programming languages give a negative result for a % n? Many languages implement the % operator as a remainder that keeps the sign of a. For example, in some environments −7 % 5 = −2. Mathematically, however, it is often more convenient to use the Euclidean remainder 3. This tool uses the mathematical convention so that modular identities and congruence rules behave as expected. How large can the numbers be? The limit is essentially the range of standard JavaScript integers. For everyday modular arithmetic, cryptography exercises, and classroom examples, this range is more than sufficient. Extremely large inputs may lose precision due to the underlying floating-point representation. Can I export the modulo table? The batch table is designed for quick on-screen inspection. To export results, you can copy the table rows into a spreadsheet or text editor and convert them into CSV or another format suitable for your workflow. Frequently Asked Questions How should I interpret the quotient and remainder? The quotient q tells you how many whole copies of n fit inside a, while the remainder r tells you what is left over. Together they satisfy a = q·n + r with 0 ≤ r < n. The modulo result is simply this remainder r. Why does the calculator sometimes show large negative quotients? When a is negative, the quotient q may also be negative to keep the remainder r in the interval [0, n). This is normal in Euclidean division and does not affect the correctness of the modulo result. Is this calculator suitable for modular arithmetic in number theory? Yes. Because it enforces the Euclidean definition of remainder, the results are consistent with standard number theory textbooks and proofs. You can safely use it to check modular equations and congruences. Related Core Math & Algebra tools Explore more tools that support modular arithmetic, number theory, and core algebra. Set theory Interval notation Base converter Exponent Fibonacci number Prime number Euclidean algorithm Continued fraction Dice roll probability Vector Pythagorean theorem SOHCAHTOA (trigonometry) Grade calculator All Math & Conversions tools Modulo quick reference Euclidean division a = qn + r, 0 ≤ r < n Modulo a mod n = r Negative a r = a − n⌊a/n⌋ Congruence a ≡ b (mod n) ⇔ n | (a − b) Canonical range 0 ≤ a mod n < n Use these identities along with the calculator to verify hand calculations and better understand modular arithmetic.

Home › Math & Conversions › Core Math & Algebra › Monte Carlo Simulation Monte Carlo Simulation Calculator Run Monte Carlo simulations directly in your browser. Choose a probability distribution, define the number of runs, and obtain an empirical distribution of outcomes with summary statistics, percentiles and the probability of exceeding a threshold. This tool is suitable for statistics students, engineers, analysts and decision-makers who need a quick way to explore uncertainty and risk without installing any software. 1. Choose distribution and parameters Distribution type Normal (Gaussian) Uniform Triangular Custom discrete Start with a simple distribution. For more complex models you can approximate the outcome as a single random variable and simulate it here. Mean (μ) Standard deviation (σ) σ must be positive. Many finance and engineering models are approximated with a normal distribution. Minimum (a) Maximum (b) Assumes all values between a and b are equally likely. Minimum (a) Most likely (m) Maximum (b) Must satisfy a <= m <= b. Common in project management (optimistic, most likely, pessimistic). Custom discrete values and probabilities 10:0.2 20:0.5 30:0.3 Probabilities must be non-negative and sum to 1 (within rounding). Values can be any real numbers. 2. Simulation settings Number of simulations Up to 200,000 runs per execution. Threshold (optional) If set, the tool will estimate P(X ≥ threshold). Decimals in output Controls rounding in the statistics table. Run simulation Clear results Pure client-side – no data uploaded Monte Carlo results Summary statistics Simulations (n) Mean Standard deviation Minimum Maximum Median (50th pct) 5th percentile 95th percentile Threshold P(X ≥ threshold) Simulation settings used Results are empirical: repeat the simulation (or increase n) to see how estimates stabilize. Approximate histogram (10 bins) Copy stats as CSV Bin range Count Relative freq. Cumulative freq. Histogram bins are computed from the simulated minimum and maximum using equal-width bins. For publication-quality plots, export the CSV statistics and use a dedicated plotting tool. What is a Monte Carlo simulation? A Monte Carlo simulation is a numerical method that uses repeated random sampling to estimate the behavior of a system with uncertainty. Instead of solving a problem analytically, you model the uncertain inputs as random variables and simulate many possible scenarios. After many runs, you get an empirical approximation of the distribution of outcomes: averages, percentiles, and probabilities. This is extremely useful when the model is too complex for closed-form formulas, or when you want to visualize risk in a way that stakeholders can understand. Basic Monte Carlo procedure Define a model \( Y = g(X_1, X_2, \dots, X_k) \) where \( X_i \) are random inputs. Choose probability distributions for each \( X_i \) (e.g. normal, uniform, triangular). For each simulation run \( j = 1, \dots, N \): Sample \( x_1^{(j)}, x_2^{(j)}, \dots, x_k^{(j)} \) from their distributions. Compute the outcome \( y^{(j)} = g(x_1^{(j)}, \dots, x_k^{(j)}) \). Use the simulated outcomes \( y^{(1)}, \dots, y^{(N)} \) to estimate: Mean and standard deviation. Percentiles (e.g. 5th, 50th, 95th). Probabilities of events, e.g. \( \mathbb{P}(Y \geq y_0) \). Supported distributions in this calculator Normal (Gaussian) distribution Defined by a mean \( \mu \) and standard deviation \( \sigma \), with density \[ f(x) = \frac{1}{\sigma \sqrt{2\pi}} \exp\left( -\frac{(x-\mu)^2}{2\sigma^2} \right). \] This is widely used for modelling measurement errors, aggregated effects and approximate returns. Uniform distribution All values in \([a,b]\) are equally likely. This is useful when you only know a range, not a shape. Triangular distribution Defined by minimum \( a \), most-likely value \( m \) and maximum \( b \). It approximates expert estimates of “optimistic–most likely–pessimistic” scenarios and is common in project management and cost estimation. Custom discrete distribution You specify a finite set of values \( x_i \) and probabilities \( p_i \) such that \( p_i \ge 0 \) and \( \sum_i p_i = 1 \). This covers scenarios like demand levels, failure modes, or aggregated outcomes from more complex models. Monte Carlo simulation – FAQ How accurate is a Monte Carlo estimate? + The accuracy improves roughly with \( 1/\sqrt{N} \), where \( N \) is the number of simulations. Doubling the number of runs reduces the standard error by about 30%. For many practical decisions, a few tens of thousands of runs are more than enough, but extremely tight confidence intervals may require more. Can I use this for financial risk analysis? + Yes, as a didactic and exploratory tool. For example, you can model annual return as a normal distribution and simulate terminal wealth. However, real financial modelling often requires correlated variables, time-series structure and more complex distributions than this simple one-variable simulator can provide. Why do results change each time I run the simulation? + The simulation uses pseudo-random numbers, so each run samples a different set of scenarios. This is expected behavior in Monte Carlo methods. If you need more stable estimates, increase the number of runs or average results from multiple runs. What if my model has several uncertain inputs? + This calculator focuses on a single aggregate random variable. In practice, you can model each input separately in a spreadsheet or script, compute the resulting outcome for each scenario, and then import the resulting values as a custom discrete distribution, or approximate them with one of the supported distributions. Core Math & Algebra tools G-test (likelihood-ratio test) FOIL Method Calculator Graph Adjacency Matrix Speed, Distance, Time Friedman Test Traveling Salesman Problem (TSP) Stem and Leaf Plot Generator Bisection Method Monte Carlo Simulation – you are here Lottery Odds Calculator Dot Product Matrix Determinant Partial Fraction Decomposition Trapezoid Area Combination and Permutation More statistics & risk tools Percent Error Calculator Effect Size (Cohen’s d) Kruskal–Wallis Test Spearman’s Rank Correlation

Home › Math & Conversions › Core Math & Algebra › Partial Fraction Decomposition Calculator Partial Fraction Decomposition Calculator This partial fraction decomposition calculator rewrites a rational function \( \frac{P(x)}{Q(x)} \) as a sum of simpler fractions with factored denominators . It is aimed at students and professionals working with integration, Laplace transforms, and algebraic manipulation of rational functions. Enter a numerator polynomial \( P(x) \) and a denominator in factored form built from distinct linear factors, such as \( (x-1)(x+2)(2x-3) \). The tool performs polynomial long division if needed and then solves for the partial fraction coefficients using a linear system of equations. 1. Define the rational function Numerator polynomial \( P(x) \) Use x as the variable, ^ for powers, and standard algebraic notation: examples: x^2+3x-4 , -0.5x^3 + 2x . Denominator in factored form \( Q(x) \) Enter a product of distinct linear factors of the form (a x + b) . Examples: (x-1)(x+2) , (2x+3)(x-4)(x+5) . Repeated factors like (x-1)^2 and quadratic factors (x^2+1) are not supported in this version. Decimal places Example: (2x+3)/[(x-1)(x+2)] Example: improper (x^2+1)/[(x-1)(x+1)] Clear inputs Decompose Clear results Requires distinct linear factors in the denominator 2. Results Decomposition overview Original rational function: Expanded denominator \( Q(x) \): Classification: Final form Polynomial part \( Q(x) \): Proper fraction remainder: Partial fractions (plain text): Partial fractions (TeX): TeX form can be pasted into LaTeX documents or CAS systems that accept LaTeX input. Coefficients summary Coefficients are rounded to the selected number of decimal places for display. Internally, the linear system is solved in double precision. Step-by-step breakdown Copy as plain text Partial fraction decomposition – core idea Given a rational function \[ \frac{P(x)}{Q(x)}, \] where \( P(x) \) and \( Q(x) \) are polynomials and \( \deg P < \deg Q \), a partial fraction decomposition rewrites it as a sum of simpler fractions whose denominators are factors of \( Q(x) \). In the simplest case where the denominator factors into distinct linear factors , \[ Q(x) = (x - r_1)(x - r_2)\cdots(x - r_n), \] we search for constants \( A_1,\dots,A_n \) such that \[ \frac{P(x)}{Q(x)} = \frac{A_1}{x - r_1} + \frac{A_2}{x - r_2} + \cdots + \frac{A_n}{x - r_n}. \] From polynomial long division to partial fractions If \( \deg P \geq \deg Q \), the rational function is improper and we start with polynomial long division: \[ \frac{P(x)}{Q(x)} = S(x) + \frac{R(x)}{Q(x)}, \] where \( S(x) \) is a polynomial (the quotient ) and \( R(x) \) is the remainder satisfying \( \deg R < \deg Q \). Only the proper fraction \( R(x)/Q(x) \) needs to be decomposed into partial fractions. Solving for the coefficients of the partial fractions Suppose the denominator is factored into distinct linear factors: \[ Q(x) = (a_1 x + b_1)(a_2 x + b_2)\cdots(a_n x + b_n). \] The calculator assumes a decomposition of the form \[ \frac{R(x)}{Q(x)} = \frac{A_1}{a_1 x + b_1} + \frac{A_2}{a_2 x + b_2} + \cdots + \frac{A_n}{a_n x + b_n}. \] Multiplying both sides by \( Q(x) \) we obtain a polynomial identity: \[ R(x) = A_1 \prod_{j\neq 1}(a_j x + b_j) + A_2 \prod_{j\neq 2}(a_j x + b_j) + \cdots + A_n \prod_{j\neq n}(a_j x + b_j). \] Expanding each product on the right-hand side and matching coefficients of powers of \( x \) yields a linear system in the unknowns \( A_1,\dots,A_n \). The calculator solves this system using Gaussian elimination, which is numerically stable for moderate degrees. Typical use cases Integration : partial fractions reduce rational integrals to sums of basic integrals of the forms \( \int \frac{1}{x-a}\,dx \) and \( \int \frac{1}{(x-a)^n}\,dx \). Laplace transform inversion : decomposing rational transfer functions into elementary terms whose inverse Laplace transforms are tabulated. Control and signal processing : analyzing poles of transfer functions and breaking them into contributions of simple first-order components. Algebraic simplification : rewriting expressions to isolate singularities or analyze asymptotic behaviour near poles. Limitations of this calculator Only distinct linear factors in the denominator are supported: terms like \( (x-1)^2 \) or \( (x^2+1) \) are intentionally rejected with a clear warning. All coefficients are treated as real numbers; complex partial fractions are not handled. For better numerical stability, choose moderate coefficient sizes and avoid highly ill-conditioned factorizations if exact arithmetic is important. Partial fraction decomposition – FAQ Why do I need the denominator in factored form? + Factoring arbitrary polynomials is a non-trivial task and can introduce numerical or symbolic ambiguities. By asking you to provide the denominator as a product of linear factors, the calculator can focus on the partial fraction step itself and give you a transparent, step-by-step solution based on a clean linear system. What happens if the degree of the numerator is too large? + If \( \deg P \geq \deg Q \), the calculator performs polynomial long division to split the function into a polynomial part plus a proper fraction. Only the proper fraction is decomposed into partial fractions. The polynomial part is displayed explicitly. Can I use this as a teaching aid in class? + Yes. The step-by-step breakdown shows the expanded denominator, the long division (when needed), and the linear system used to solve for the coefficients. You can copy the full working as plain text or TeX and adapt it to slides or handouts. Core Math & Algebra tools Bisection Method Monte Carlo Simulation Lottery Odds Dot Product Matrix Determinant Partial Fraction Decomposition – you are here Trapezoid Area Combination and Permutation Torus Volume Matrix Transpose Surface Area Related calculus tools Integral Calculator Derivative Calculator Limit Calculator

Home › Math & Conversions › Core Math & Algebra › Percent Error Calculator Percent Error Calculator Use this percent error calculator to compare a measured (observed) value with the true (accepted) value. Get instant absolute and signed percent error, absolute error and relative error, plus a clear step-by-step explanation for lab reports, QA documentation and statistics homework. Calculate percent error True

Home › Math & Conversions › Core Math & Algebra › Perfect Number Calculator Perfect Number Calculator – Perfect, Abundant, or Deficient? Enter a positive integer to check if it is a perfect number and see its proper divisors, divisor sum, and classification as perfect, abundant, or deficient. Optionally search for all perfect numbers in a range. Built for number theory students, teachers, and curious learners who want transparent, step-by-step divisor analysis. Perfect number checker Divisor-based classification Choose a mode, enter your integer(s), and click Calculate . The tool computes proper divisors, their sum, and a clear classification. Mode Single number check Scan a range for perfect numbers Integer n Enter a positive integer. For very large n the computation may take longer; this tool is optimized for classroom-sized examples. Quick examples 6 (perfect) 28 (perfect) 12 (abundant) 8 (deficient) Upper bound N N between 2 and 1,000,000. The algorithm uses an O(√n) divisor search for each integer up to N. Range scan is for exploration and teaching. For very large searches, use theoretical results on perfect numbers or specialized software; this tool intentionally prioritizes readability and deterministic output over extreme performance. Calculate Clear Divisor search up to ⌊√n⌋ with symmetry (d, n/d) for efficiency. Results, divisor lists, and classifications will appear here after you click Calculate . What is a perfect number? A positive integer \(n\) is called a perfect number if it equals the sum of its proper divisors , that is, all positive divisors less than \(n\) itself. Formally, if \(\sigma(n)\) is the sum of all positive divisors of \(n\), then \(n\) is perfect when \[ \sigma(n) - n = n \quad\Longleftrightarrow\quad \sum_{\substack{d \mid n\\ d < n}} d = n. \] The first perfect numbers are \(6, 28, 496, 8128,\dots\). For example, the proper divisors of 28 are \(1, 2, 4, 7, 14\), and \(1 + 2 + 4 + 7 + 14 = 28\). Perfect, abundant, and deficient numbers The same divisor-sum test also gives two related classifications: Perfect : \(\sum_{d \mid n,\, d < n} d = n\). Abundant : \(\sum_{d \mid n,\, d < n} d > n\) (the number has “too much” divisor mass). Deficient : \(\sum_{d \mid n,\, d < n} d < n\) (the divisor sum falls short of the number). Many everyday integers such as 8, 10, and 11 are deficient; abundant numbers like 12 and 18 appear regularly; perfect numbers are extremely rare. Even perfect numbers and Mersenne primes All known perfect numbers are even and have a special form discovered by Euclid and Euler: \[ n = 2^{p-1}\,\bigl(2^p - 1\bigr), \] where \(2^p - 1\) is a Mersenne prime , a prime number of the form \(2^p - 1\). Every time a new Mersenne prime \(2^p - 1\) is found, it generates a new even perfect number. Modern record perfect numbers are discovered using large-scale distributed computing projects. Do odd perfect numbers exist? No one knows. Despite centuries of work, no odd perfect number has ever been found . At the same time, there is no proof that odd perfect numbers cannot exist; it remains a deep open problem in number theory. How the calculator finds proper divisors For a given \(n\), the calculator performs an efficient divisor search: It always includes 1 as a proper divisor (for \(n > 1\)). It loops over integers \(d\) from 2 up to \(\lfloor\sqrt{n}\rfloor\). Whenever \(d\) divides \(n\), it adds both \(d\) and \(n/d\) (when distinct) to the divisor list. It removes \(n\) itself from the list, sorts the remaining divisors, and sums them. This algorithm runs in \(O(\sqrt{n})\) time, which is efficient for the typical ranges used in teaching and exploratory work. Perfect number calculator – FAQ Is 1 a perfect number? + No. The proper divisors of 1 are usually taken to be the empty set, so their sum is 0, which is not equal to 1. For this reason, 1 is classified as deficient, not perfect. The smallest perfect number is 6. Why does the range feature stop at 1,000,000? + The limit keeps the tool fast and responsive in a standard web browser while still covering all classic textbook examples. The divisor search is deliberately implemented in a clear, deterministic way rather than using extremely low-level optimizations. For very large searches, dedicated number theory software or distributed projects are more appropriate. Can this tool prove that a number is not perfect? + For any specific integer n in the supported range, the tool computes all proper divisors and their sum. If the sum does not equal n, then n is not perfect—this is a direct check of the definition, not a heuristic. However, proving that no perfect numbers exist beyond a certain range is a much deeper theoretical question and cannot be settled by finite computation alone. Core Math & Algebra tools Modular Arithmetic Equation Solver Circular Segment Area Trig Identity Explorer Hyperbolic Function Explorer Linear Algebra Toolbox Trapezoidal Rule Calculator Perfect Number Calculator (you are here) Chinese Remainder Theorem Set Theory Basics Number theory & probability Base Converter Interval Notation Helper Pythagorean Theorem Convolution Explorer Dice Roll Probability Professional use note This perfect number calculator is ideal for teaching, exam preparation, and exploratory number theory. For research-level work on perfect numbers or Mersenne primes, consult specialized literature and software that implements advanced primality tests and large-integer arithmetic.

Home › Math & Conversions › Core Math & Algebra › Prime number Prime Number Calculator Check whether a number is prime, factor composite integers, and list primes in a range – with clear explanations and number theory context. Core Math & Algebra Interactive prime number tool Use this calculator to test if a number is prime, see its prime factorization if composite, or generate a list of primes within a given interval. Designed for students, teachers, and anyone working with number theory or cryptography basics. Prime check & factorization Prime list in range Integer to test Enter a positive integer ≥ 2. For very large values, computation may take a bit longer. Quick tips Even numbers > 2 are not prime. Numbers ending in 5 and > 5 are not prime. You only need to test divisors up to \(\sqrt{n}\). Check primality Clear Load example: large composite Result summary – Classification Prime factorization Divisibility reasoning Start of range End of range Guidelines For performance and readability, keep the range length ≤ 100 000. The tool will truncate the displayed list after 1 000 primes while still reporting the full count. List primes in range Clear Primes in range Summary Prime list What is a prime number? A prime number is a natural number greater than 1 that has exactly two positive divisors: 1 and itself. Numbers greater than 1 that are not prime are called composite , because they can be written as a product of smaller positive integers. The first few prime numbers are: \[ 2,\;3,\;5,\;7,\;11,\;13,\;17,\;19,\;23,\;29,\dots \] Note that 2 is the only even prime. Every other even number is divisible by 2 and is therefore composite. Prime factorization and the building blocks of integers A key fact from elementary number theory is the Fundamental Theorem of Arithmetic : Every integer \(n > 1\) is either prime or can be written as a product of prime numbers in a way that is unique up to the order of the factors. For example: \(60 = 2^2 \cdot 3 \cdot 5\) \(84 = 2^2 \cdot 3 \cdot 7\) \(131\) is prime, so its prime factorization is just \(131\) itself. The factorization returned by the calculator is written in a standard prime-power format, such as \(2^3 \cdot 7\), which is convenient for later computations with greatest common divisors, least common multiples, and modular arithmetic. How to test if a number is prime For a given integer \(n \ge 2\), a simple primality test is: If \(n = 2\) or \(n = 3\), it is prime. If \(n\) is divisible by 2 or 3 and greater than 3, it is composite. Otherwise, test divisibility by all integers of the form \(6k \pm 1\) up to \(\sqrt{n}\). If none of these integers divides \(n\), then \(n\) is prime. This strategy is efficient enough for educational and everyday use on the range of values supported by this calculator. Prime number density (how often primes occur) Intuitively, prime numbers become less frequent as numbers grow larger, but they never run out. The Prime Number Theorem states that the number of primes up to \(x\), usually denoted \(\pi(x)\), is asymptotically \(\displaystyle \pi(x) \sim \frac{x}{\log x}\) This means that around a large number \(x\), the probability that a random integer is prime is roughly \(\frac{1}{\log x}\). The summary shown by the prime-range tool reports the empirical density of primes in your chosen interval, which you can compare with this heuristic. Prime number – FAQ Why is 1 not considered a prime number? If 1 were prime, it would break the uniqueness part of the Fundamental Theorem of Arithmetic, because you could insert extra factors of 1 into any factorization. By definition, primes have exactly two positive divisors, which rules out 1 (it has only one positive divisor: itself). Are negative numbers prime? In elementary number theory, primes are usually restricted to positive integers greater than 1. For negative integers, one can talk about units and associate each negative integer with its positive counterpart, but the standard definition of a prime number refers to natural numbers only. Does this tool handle extremely large primes like those used in cryptography? No. This calculator focuses on pedagogical ranges: classroom-sized numbers, contest problems, and quick checks. Modern cryptographic applications rely on specialized algorithms such as probabilistic primality tests and large integer arithmetic libraries, which are designed specifically for very large integers. What is a twin prime? A twin prime pair is a pair of primes that differ by 2, such as (3,5), (5,7), (11,13), or (17,19). It is an open problem in mathematics whether there are infinitely many twin primes. How can I use this calculator in teaching or exams? In teaching, the tool is helpful for quickly checking answers and exploring patterns (for example, how primes are distributed in different ranges). In exams, students are usually expected to show their own reasoning and divisibility tests; the calculator is best used as a companion for practice and verification before or after formal assessments. Frequently Asked Questions What are some quick mental checks for non-prime numbers? Several shortcuts help you discard many composite numbers instantly: any even number greater than 2 is not prime; any number ending in 5 and greater than 5 is divisible by 5; and numbers whose digits sum to a multiple of 3 are divisible by 3. Combining these rules can quickly filter out common composites before more detailed tests. Should I always find the full factorization to decide if a number is prime? No. To determine if a number is prime, it is sufficient to show that no prime up to its square root divides it. Full factorization is needed only when you want the exact prime factors of a composite number, which this calculator provides when feasible. How can I use prime factorization in practical problems? Prime factorization is a powerful tool for computing greatest common divisors, least common multiples, simplifying fractions, working with modular arithmetic, and understanding divisibility constraints in combinatorics and probability problems. Related Core Math & Algebra tools Deepen your number theory and algebra skills with these companion tools. Set theory basics Interval notation Base converter Exponent rules Fibonacci number Continued fraction Dice roll probability Pythagorean theorem SOHCAHTOA (trigonometry) Convolution All Math & Conversions tools Prime number checklist Confirm \(n\) is an integer and \(n \ge 2\). Eliminate trivial composites: check divisibility by 2, 3, and 5. Test divisors up to \(\sqrt{n}\) using the \(6k\pm1\) pattern. Record prime factorization if any divisor is found. For large or security-critical values, use specialized primality testing libraries.

Home › Math & Conversions › Core Math & Algebra › Pythagorean theorem Pythagorean Theorem Calculator Find the hypotenuse or a missing leg of a right triangle with \(a^2 + b^2 = c^2\). Get step-by-step work, area, perimeter and a quick diagram. Core Math & Algebra Interactive right triangle calculator Choose which side you want to solve for, enter the known sides, and the calculator will: Check that the triangle can be right-angled. Compute the missing side using \(a^2 + b^2 = c^2\). Report area and perimeter. Show clear, step-by-step working. 1. What do you want to calculate? Hypotenuse \(c\) Leg \(a\) Leg \(b\) Leg \(a\) Leave blank if \(a\) is the unknown side. Leg \(b\) Leave blank if \(b\) is the unknown side. Hypotenuse \(c\) Leave blank if \(c\) is the unknown side. Units (optional) All sides must use the same unit. Decimal places 0 (nearest whole) 1 decimal 2 decimals 3 decimals 4 decimals 6 decimals Calculate Clear Load 3–4–5 example Results Triangle geometry Check: does it form a right triangle? Step-by-step working Right triangle diagram b (base) a (height) c (hypotenuse) Side \(c\) is always the hypotenuse, opposite the right angle. Sides \(a\) and \(b\) are the legs. The Pythagorean theorem in a nutshell The Pythagorean theorem describes the relationship between the sides of a right triangle. If \(a\) and \(b\) are the legs and \(c\) is the hypotenuse (the side opposite the right angle), then \(a^2 + b^2 = c^2\) This means that the square of the hypotenuse is equal to the sum of the squares of the other two sides. It is one of the most widely used results in geometry, physics, engineering, navigation and everyday construction. Solving for different sides Depending on which side is unknown, you can rearrange the formula: To find the hypotenuse \(c\): \(c = \sqrt{a^2 + b^2}\). To find a leg \(a\): \(a = \sqrt{c^2 - b^2}\). To find the other leg \(b\): \(b = \sqrt{c^2 - a^2}\). In all cases, \(c\) must be the longest side; otherwise, the triangle cannot be right-angled. Worked example (3–4–5 triangle) Suppose you have a right triangle with legs \(a = 3\) and \(b = 4\). The hypotenuse is: \(c = \sqrt{a^2 + b^2} = \sqrt{3^2 + 4^2} = \sqrt{9 + 16} = \sqrt{25} = 5.\) Area and perimeter of a right triangle Once you know all three sides, you can immediately compute: Area : \(\dfrac{1}{2} a b\). Perimeter : \(P = a + b + c\). The calculator shows these values automatically, using the same units as your inputs. When can you use the Pythagorean theorem? The Pythagorean theorem is only valid for right triangles (one angle is exactly \(90^\circ\)). Common uses include: Finding a ladder length safely reaching a given height. Checking whether a corner is square in construction. Computing the diagonal of a rectangle or screen. Distance between two points on a coordinate grid. For non-right triangles, you generally need the law of cosines , which reduces to the Pythagorean theorem when the angle is \(90^\circ\). Connection with the distance formula In coordinate geometry, the distance between points \((x_1, y_1)\) and \((x_2, y_2)\) is \(d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}.\) This is just the Pythagorean theorem in disguise: the horizontal and vertical differences are the legs of a right triangle, and the distance is the hypotenuse. Pythagorean theorem – FAQ What is the Pythagorean theorem? It is a statement about the lengths of the sides of a right triangle: the square of the hypotenuse equals the sum of the squares of the legs, \(a^2 + b^2 = c^2\). How do I know which side is the hypotenuse? The hypotenuse is always the side opposite the right angle and is always the longest side of a right triangle. In the calculator and formulas on this page, the hypotenuse is always called \(c\). What units does the calculator use? You can use any unit of length (meters, centimeters, inches, feet, etc.), as long as all sides use the same unit. The calculator carries the unit through to the results for area and perimeter. Can the Pythagorean theorem be used in 3D? Yes, but you apply it step by step. For example, to find the space diagonal of a box with sides \(x\), \(y\), and \(z\), first use the theorem in the base to get the diagonal \(d = \sqrt{x^2 + y^2}\), then use it again with \(d\) and \(z\) to get the full diagonal \(\sqrt{x^2 + y^2 + z^2}\). What if my numbers don’t satisfy a² + b² = c² exactly? In real measurements you usually have rounding and small errors. The calculator reports how closely \(a^2 + b^2\) matches \(c^2\) and flags impossible combinations (for example, if a leg is longer than the hypotenuse). Frequently Asked Questions How accurate is the Pythagorean theorem calculator? The calculator uses standard floating-point arithmetic with user-selectable rounding. For classroom work and most engineering or construction tasks, this is more than sufficient. If you need very high precision, increase the number of decimal places or work symbolically. Can this tool prove that a triangle is right-angled? If you already know all three side lengths, you can check whether they satisfy \(a^2 + b^2 \approx c^2\). If the equality holds to within measurement error, the triangle is right-angled. The calculator reports the difference between \(a^2 + b^2\) and \(c^2\) so you can make an informed judgement. What are classic Pythagorean triples? Pythagorean triples are integer solutions to \(a^2 + b^2 = c^2\), such as (3, 4, 5), (5, 12, 13) or (8, 15, 17). They are especially useful when you want exact lengths without rounding, for example in carpentry or layout work. Related Core Math & Algebra tools Explore more foundational math tools that pair naturally with the Pythagorean theorem. Set theory Base converter Interval notation Probability Chinese remainder theorem Perfect number Continued fraction Exponent Fibonacci number All Math & Conversions tools Quick checklist before you trust the result Confirm that the triangle is intended to be right-angled. Make sure all sides are in the same unit (e.g., all in cm). Check that the hypotenuse is the largest side. Compare \(a^2 + b^2\) with \(c^2\); large differences indicate an input error. Round the final answer appropriately for your application.

Home › Math & Conversions › Core Math & Algebra › SOHCAHTOA SOHCAHTOA Calculator (Sine, Cosine & Tangent) Learn and apply SOHCAHTOA: compute sin, cos, and tan for any angle, or solve a right triangle from one angle and one side with clear step-by-step explanations. Core Math & Algebra Interactive SOHCAHTOA tool Use this panel in two ways: Trig ratios from angle gives you \(\sin\theta\), \(\cos\theta\), and \(\tan\theta\) directly. Right triangle solver applies SOHCAHTOA to find missing side lengths from one acute angle and one known side of a right triangle. Trig ratios from angle Right triangle (SOHCAHTOA) Switch between pure trig ratios and geometric right-triangle mode. Angle \(\theta\) Enter a number; you can also type common angles like 30, 45, 60. Angle unit degrees (°) radians (rad) Switch between degrees and radians; the calculator converts internally. Quick angles 30° 45° 60° 90° Common special angles for exact SOHCAHTOA triangles. Compute sin, cos, tan Clear Trig ratios at \(\theta\) In this mode, we assume a right triangle with one 90° angle. The angle \(\theta\) is one of the acute angles (\(0^\circ < \theta < 90^\circ\)). SOHCAHTOA applies to the sides relative to this angle. Acute angle \(\theta\) Must be between 0° and 90° (or 0 and \(\pi/2\) radians). Angle unit degrees (°) radians (rad) Choose the unit you are using for \(\theta\). Side length unit (optional label) Used only for display; does not affect calculations. Known side (relative to \(\theta\)) Hypotenuse Opposite side Adjacent side Known side length Must be a positive number; same unit as any other sides. Solve right triangle Clear Example: 3–4–5 triangle Right triangle solution Side lengths Trig ratios at \(\theta\) SOHCAHTOA summary What is SOHCAHTOA? SOHCAHTOA is a compact mnemonic for remembering the definitions of the three primary trigonometric ratios in a right triangle, relative to an acute angle \(\theta\): SOH: \(\displaystyle \sin\theta = \frac{\text{Opposite}}{\text{Hypotenuse}}\) CAH: \(\displaystyle \cos\theta = \frac{\text{Adjacent}}{\text{Hypotenuse}}\) TOA: \(\displaystyle \tan\theta = \frac{\text{Opposite}}{\text{Adjacent}}\) Here, “opposite” means the side directly across from the angle \(\theta\), “adjacent” is the side that touches \(\theta\) but is not the hypotenuse, and the “hypotenuse” is the longest side, opposite the right angle. Using SOHCAHTOA to find missing sides To use SOHCAHTOA effectively: Draw or imagine the right triangle and clearly mark the angle \(\theta\) you are using. Label each side as opposite , adjacent , or hypotenuse relative to \(\theta\). Decide which ratio uses the known side and the unknown side: SOH, CAH, or TOA. Set up the equation and solve for the unknown side length. The right-triangle panel of this calculator automates these steps once you select the known side type and enter its value. Degrees vs radians Trigonometric functions accept angles in either degrees or radians . Many school problems use degrees, while calculus and programming typically use radians. You can convert between them via \(\text{radians} = \text{degrees} \times \dfrac{\pi}{180}, \quad \text{degrees} = \text{radians} \times \dfrac{180}{\pi}.\) This calculator has a unit selector for both angle modes and performs the conversion internally, so the formulas for SOHCAHTOA stay consistent. Special SOHCAHTOA triangles Certain angles produce especially nice ratios. Two classic examples are: The 30°–60°–90° triangle , with side ratios \(1 : \sqrt{3} : 2\). For example, if the hypotenuse is 2, then the shorter leg is 1 and the longer leg is \(\sqrt{3}\). The 45°–45°–90° triangle , with side ratios \(1 : 1 : \sqrt{2}\). Both legs are equal. The quick-angle buttons use these special triangles so you can see the corresponding sin, cos, and tan values and use them as reference points. SOHCAHTOA – FAQ Can I use SOHCAHTOA on any triangle? No. SOHCAHTOA only applies to right triangles , where one angle is exactly 90°. For non-right triangles you need other tools such as the Law of Sines or the Law of Cosines. What happens if the angle is 0° or 90°? At 0° and 90°, some ratios become undefined (for example, \(\tan 90^\circ\)). The right-triangle solver therefore requires \(0^\circ < \theta < 90^\circ\). The angle-only trig mode can still evaluate sin, cos, and tan at boundary angles but will indicate when a ratio is undefined or numerically unstable. Why does \(\tan\theta = \frac{\sin\theta}{\cos\theta}\)? From SOHCAHTOA, \(\sin\theta = \frac{\text{Opp}}{\text{Hyp}}\) and \(\cos\theta = \frac{\text{Adj}}{\text{Hyp}}\). Dividing these expressions gives \(\dfrac{\sin\theta}{\cos\theta} = \dfrac{\text{Opp}/\text{Hyp}}{\text{Adj}/\text{Hyp}} = \dfrac{\text{Opp}}{\text{Adj}} = \tan\theta.\) This identity is built into the calculator checks to ensure internal consistency. How accurate is this SOHCAHTOA calculator? The tool uses double-precision floating-point arithmetic, the same standard used by scientific software and calculators. Results are rounded for display, but internal operations have more precision than most classroom or design tasks require. Frequently Asked Questions Is SOHCAHTOA valid in any quadrant? The SOHCAHTOA side definitions come from a right triangle picture, which naturally lives in the first quadrant where all sides and ratios are positive. When you extend sine, cosine, and tangent to other quadrants on the unit circle, the numerical definitions stay the same, but the signs of sin, cos, and tan change according to the quadrant. The angle-only panel includes a summary of sign patterns by quadrant. Does the calculator keep track of opposite/adjacent automatically? Yes. When you choose which side is known (opposite, adjacent, or hypotenuse) and enter the acute angle, the right-triangle solver applies the appropriate SOH, CAH, or TOA relationship and solves for the missing sides using the standard right-triangle formulas. Can I use this tool in programming or engineering work? Absolutely. The calculator mirrors the behaviour of common math libraries for sin, cos, and tan while providing a geometric interpretation via SOHCAHTOA. It is ideal for quick checks of angle conversions, right-triangle dimensions, and sign conventions before you implement formulas in code or design documents. Related trigonometry & core math tools Strengthen your trigonometry and algebra toolkit with these calculators from the same Core Math & Algebra family. Pythagorean theorem Ellipse area Fibonacci number Continued fraction Dice roll probability Exponent calculator Set theory basics Interval notation Base converter All Math & Conversions tools SOHCAHTOA quick checklist Confirm you are working with a right triangle (one angle is 90°). Select an acute reference angle \(\theta\) and label opposite, adjacent, and hypotenuse. Pick the ratio that matches the sides you know and need: SOH, CAH, or TOA. Keep track of angle units (degrees vs radians) consistently. For sanity checks, verify that \(\sin^2\theta + \cos^2\theta \approx 1\) and \(\tan\theta \approx \sin\theta

Home › Math & Conversions › Core Math & Algebra › Speed, distance, time Speed, Distance, Time Calculator Use this calculator to solve for speed , distance or time from the basic relationship \(v = d

Home › Math & Conversions › Core Math & Algebra › Stem and Leaf Plot Generator Stem and Leaf Plot Generator Create a stem and leaf plot from raw data in seconds. Paste your numbers, choose how to split stems and leaves, and get a clean text-based display that preserves every value. This tool is ideal for GCSE, AP Statistics, college intro stats, and quick exploratory data analysis when a back-of-the-envelope visualization is all you need. 1. Enter your data Data values 12 15 19 21 21 23 25 27 29 31 Use spaces, commas or line breaks as separators. Only numeric values (integers or decimals) are allowed. 2. Configure stems and leaves Data type Integer data (leaf = ones digit) Example: 12 → 1 | 2 , 37 → 3 | 7 . 1 decimal place (leaf = first decimal) Example: 12.3 → 12 | 3 , 8.7 → 8 | 7 . Display options Split stems (0–4 and 5–9) When checked, each stem can appear on two rows: one for leaves 0–4 and one for leaves 5–9. Leaf order Ascending (standard) Descending Generate plot Clear n ≤ 300 values recommended Stem and leaf plot Summary statistics Count (n) Min Max Mean Median Mode(s) Settings used Stem and leaf table Copy as text Stem Leaves Remember: the stem-and-leaf plot preserves each data value. You can reconstruct the original dataset from the plot by combining stems and leaves. What is a stem and leaf plot? A stem and leaf plot (or stem-and-leaf display) is a compact way to visualize quantitative data using text only. Each observation is split into: Stem – all but the last digit (or all but the first decimal digit). Leaf – the last digit (or the first decimal digit). For example, with integer data 12, 15, 19, 21, 21, 23, 25, 27, 29, 31 we can write: 1 | 2 5 9 2 | 1 1 3 5 7 9 3 | 1 Here, the stems 1, 2, 3 represent the tens (10s, 20s, 30s) and the leaves represent the ones digit. The resulting display shows the distribution’s shape, clusters and possible outliers while keeping every data value visible. How this calculator builds the plot 1. Parsing and cleaning data The generator accepts values separated by spaces, commas or line breaks. It removes empty entries, converts decimal commas to dots (e.g. 12,5 → 12.5) and rejects any non-numeric items. 2. Splitting into stems and leaves Two main modes are supported: Integer data : stem = floor(x

Home › Math & Conversions › Core Math & Algebra › Trapezoid Area Calculator Trapezoid Area Calculator This trapezoid area calculator lets you compute the area of a trapezoid in several ways: from bases and height , from bases and equal leg for an isosceles trapezoid, or directly from the coordinates of the four vertices . It also reports perimeter (when side lengths are known), the midsegment length , and the height (if it is derived instead of entered). Results are designed for both students and professionals who need fast and reliable geometric calculations. 1. Enter trapezoid dimensions Choose input method Bases + height Isosceles (bases + leg) 4 vertices (coordinates) Linear unit meters (m) centimeters (cm) millimeters (mm) inches (in) feet (ft) Decimal places Provide both parallel bases \( b_1 \) and \( b_2 \), the height \( h \), and optionally the non-parallel sides \( c \) and \( d \) for the perimeter. Base 1 \( b_1 \) Base 2 \( b_2 \) Height \( h \) Side \( c \) (optional) Side \( d \) (optional) For an isosceles trapezoid , the two non-parallel sides are equal. Provide the two bases and the leg length. The calculator derives the height using Pythagoras and then computes area and perimeter. Base 1 \( b_1 \) Base 2 \( b_2 \) Leg (equal side) \( l \) Enter the coordinates of the four vertices of the trapezoid in order around the shape (clockwise or counterclockwise). The tool uses the shoelace formula to compute area and also returns side lengths. Vertex A Vertex B Vertex C Vertex D Calculate Clear All dimensions must use the same linear unit 2. Results Geometric summary Method: Key inputs: Classification: Area & core metrics Area: Height: Midsegment length: Perimeter: Side lengths (when available) Formula used Trapezoid area – main formulas A trapezoid (or trapezium in some regions) is a quadrilateral with at least one pair of parallel sides. Those parallel sides are called the bases , usually denoted \( b_1 \) and \( b_2 \). The distance between them is the height \( h \). The fundamental area formula is \[ A = \frac{(b_1 + b_2)\,h}{2}. \] Intuitively, this is the area of a rectangle with height \( h \) and base equal to the average of the two bases: \[ A = \overline{m}\,h, \quad \overline{m} = \frac{b_1 + b_2}{2}. \] Isosceles trapezoid – bases and equal leg In an isosceles trapezoid the non-parallel sides (legs) are equal: \( c = d = l \). If we know the two bases \( b_1 \leq b_2 \) and the leg \( l \), we can recover the height using Pythagoras. The horizontal offset at each side is \[ x = \frac{b_2 - b_1}{2}, \] and the height is obtained from the right triangle: \[ h = \sqrt{l^2 - x^2}, \] assuming \( l \geq x \). Then the area follows from the main formula above. The perimeter is \[ P = b_1 + b_2 + 2l. \] Coordinates method – shoelace formula If the vertices \( A(x_1,y_1), B(x_2,y_2), C(x_3,y_3), D(x_4,y_4) \) are known in order around the trapezoid, we can compute the area using the shoelace formula : \[ A = \frac{1}{2}\left| x_1y_2 + x_2y_3 + x_3y_4 + x_4y_1 - (y_1x_2 + y_2x_3 + y_3x_4 + y_4x_1) \right|. \] The calculator also computes each side length using the Euclidean distance \[ \text{side length} = \sqrt{(x_i - x_j)^2 + (y_i - y_j)^2}, \] and reports the perimeter whenever all four sides are well-defined. Worked example – bases and height Suppose a trapezoid has bases \( b_1 = 8\;\text{m} \), \( b_2 = 14\;\text{m} \), and height \( h = 5\;\text{m} \). Compute the midsegment: \( \overline{m} = (8 + 14)/2 = 11 \;\text{m} \). Compute the area: \( A = \overline{m}\,h = 11 \cdot 5 = 55\;\text{m}^2 \). Entering these values into the calculator with unit meters returns an area of \( 55\;\text{m}^2 \), plus the chosen extra metrics if side lengths are supplied. Trapezoid geometry – FAQ What is the midsegment of a trapezoid? + The midsegment (or median) of a trapezoid is the segment that joins the midpoints of the two non-parallel sides. Its length is the average of the bases: \[ \overline{m} = \frac{b_1 + b_2}{2}. \] The area can be interpreted as the midsegment length times the height: \( A = \overline{m}\,h \). Do the non-parallel sides affect the area? + No. For a given pair of bases and height, the area is completely determined by \[ A = \frac{(b_1 + b_2)\,h}{2}. \] The non-parallel sides affect the perimeter and the overall shape but not the area, as long as the bases and the height remain fixed. How can I tell if four points form a trapezoid? + In analytic geometry, a quadrilateral is a trapezoid if at least one pair of opposite sides is parallel. You can check this by comparing slopes of sides: \[ \text{slope}(AB) = \text{slope}(CD) \] or \[ \text{slope}(BC) = \text{slope}(DA). \] This calculator focuses on area and does not fully validate whether a four-point input is strictly a trapezoid, so it is good practice to check the geometry if this matters. Core Math & Algebra tools Monte Carlo Simulation Lottery Odds Dot Product Matrix Determinant Partial Fraction Decomposition Trapezoid Area – you are here Combination and Permutation Torus Volume Matrix Transpose Surface Area Modular Exponentiation Related geometry tools Triangle Area Rectangle Area Circle Area

Home › Math & Conversions › Core Math & Algebra › Trapezoidal Rule Calculator Trapezoidal Rule Calculator – Numerical Integration Approximate definite integrals using the trapezoidal rule. Work from a function \(f(x)\) on an equally spaced grid or from tabulated data points \((x_i, f(x_i))\). Designed for calculus students, instructors, and engineers who need a transparent, step-by-step numerical integration tool. Trapezoidal rule calculator Composite & data-based trapezoidal rule Choose the calculation mode, fill in the inputs, and click Calculate . The tool reports the trapezoidal approximation, effective step size, and a detailed grid table. Calculation mode Function-based (equally spaced grid) Data points \((x_i, f(x_i))\) Lower limit a Upper limit b Number of subintervals n n ≥ 1. A larger n gives a finer grid and usually better accuracy. Function \(f(x)\) Use standard notation. The calculator automatically maps sin, cos, tan, exp, ln, log, sqrt, abs, pi, e to JavaScript’s Math.* functions and interprets ^ as a power operator. Enter pairs \((x_i, f(x_i))\). Empty rows are ignored. The tool sorts your points by x and applies the basic trapezoid formula on each interval. Number of data points: Update table Calculate Clear Numerical tolerance for diagnostics: 1e-12 The trapezoidal approximation and grid table will appear here after you click Calculate . Understanding the trapezoidal rule The trapezoidal rule is one of the simplest and most widely used numerical integration formulas. Instead of computing the exact area under the curve \(y = f(x)\), you approximate the graph with straight-line segments and sum the areas of the resulting trapezoids. 1. Basic trapezoid formula On a single interval \([x_i, x_{i+1}]\) with function values \(f(x_i)\) and \(f(x_{i+1})\), the area of the trapezoid is: \[ \int_{x_i}^{x_{i+1}} f(x)\,dx \approx \frac{x_{i+1} - x_i}{2}\,\bigl(f(x_i) + f(x_{i+1})\bigr). \] This formula works whether your data come from a function or from measurements. The only requirement is that you know the x-coordinates and the corresponding function (or measured) values. 2. Composite trapezoidal rule on an equally spaced grid If you divide \([a, b]\) into \(n\) equal subintervals of width \(h = \frac{b-a}{n}\), with grid points \(x_0 = a, x_1, \dots, x_n = b\), the composite trapezoidal rule is: \[ \int_a^b f(x)\,dx \approx \frac{h}{2} \left[f(x_0) + 2\sum_{i=1}^{n-1} f(x_i) + f(x_n)\right]. \] The function-based mode of this calculator builds exactly this grid, evaluates \(f(x_i)\) at each point, and applies the formula. 3. Error behavior If \(f\) has a continuous second derivative on \([a, b]\), the global error of the composite trapezoidal rule satisfies \[ \text{Error} \approx -\frac{(b - a)}{12} h^2\, f''(\xi) \quad \text{for some} \;\xi \in (a, b). \] This shows a quadratic dependence on the step size \(h\): halving \(h\) (doubling the number of subintervals) typically reduces the error by roughly a factor of 4, provided \(f\) is smooth enough and does not oscillate too wildly. 4. Trapezoidal rule vs. Simpson’s rule Simpson’s rule uses quadratic interpolation on pairs of subintervals and generally achieves higher accuracy (error proportional to \(h^4\)) for smooth integrands. The trapezoidal rule, however, has important advantages: It works directly with tabulated data and nonuniform x-spacing. It is easy to implement and explain in a classroom setting. It is often sufficient for engineering estimates and quick checks. In practice, you can use the trapezoidal rule to get a first estimate, then refine or cross-check with Simpson’s rule when it is available. Trapezoidal rule – FAQ How should I enter a function like sin(x) or e −x² ? + You can type expressions such as sin(x) , cos(x) , exp(-x^2) , or sqrt(1 + x^2) . The calculator automatically converts common functions to JavaScript’s Math.* equivalents and interprets ^ as a power operator. If needed, you can always write the expression explicitly, for example Math.sin(x) or Math.exp(-x*x) . What happens if the function has a discontinuity inside [a, b]? + The trapezoidal rule assumes that the function is reasonably well-behaved on the interval. If there is a jump or vertical asymptote, the approximation may be seriously misleading. In such cases you should split the integral at the discontinuity (if the improper integral makes sense) or use a method tailored for improper integrals and compare results carefully. Can this tool tell me the exact integration error? + No. The exact error depends on the second derivative of the function, which is generally unknown when you use numerical integration. The tool reports step size and grid, so you can repeat the computation with a finer grid and compare values. The difference between two successive refinements is often a good indicator of the remaining error in practical work. Core Math & Algebra tools Set Operations Arithmetic Sequence Linear Regression Modular Arithmetic Equation Solver Circular Segment Area Trig Identity Explorer Hyperbolic Function Explorer Linear Algebra Toolbox Trapezoidal Rule Calculator (you are here) Perfect Number Checker Chinese Remainder Theorem Number theory & calculus helpers Set Theory Basics Base Converter Interval Notation Helper Pythagorean Theorem Convolution Explorer Dice Roll Probability Professional use note This trapezoidal rule calculator is suited for teaching, study, and quick engineering estimates. For safety-critical or regulatory calculations, repeat the computations with independent numerical software and verify that results are stable with respect to step size and data quality.

Home › Math & Conversions › Core Math & Algebra › Traveling Salesman Problem (TSP) Traveling Salesman Problem (TSP) Calculator Build and solve small instances of the Traveling Salesman Problem (TSP) . Enter city coordinates (X, Y) or a full distance matrix, choose an exact Held–Karp solver for up to 12 cities or a nearest-neighbor + 2-opt heuristic for larger cases, and visualize the resulting tour. This tool is ideal for teaching, coursework and quick what-if analyses in operations research, logistics and graph theory. Input mode Coordinates (X, Y) Distance matrix Number of cities (n) For exact optimization, keep n ≤ 12. Rebuild table Load example City coordinates Each row is one city. Labels are assigned automatically (A, B, C, ...). Coordinates are arbitrary: you can treat them as X/Y on a plane or as projected longitude

Home › Math & Conversions › Core Math & Algebra › Trig Identity Explorer Trig Identity Explorer & Checker Browse the most important trigonometric identities in one structured cheat sheet and use the interactive checker to test your own trig identities numerically. All trigonometric functions are evaluated in radians. Use x as the variable. Trig identity checker Enter a candidate identity in terms of x . The tool evaluates both sides at several values of x (in radians) and checks whether they agree within a small numerical tolerance. Left-hand side (LHS) Right-hand side (RHS) Supported functions: sin , cos , tan , sec , csc , cot , asin , acos , atan , sqrt , abs , log , ln , exp , and constant pi . Check identity numerically Clear The numerical check report will appear here. The tool samples several values of x in radians and compares LHS(x) and RHS(x). Basic & Pythagorean Sum

Home › Math & Conversions › Core Math & Algebra › Vector Vector Calculator Compute magnitude, dot and cross products, angle between vectors, and projections in 2D or 3D, with formulas and geometric interpretation. Core Math & Algebra Interactive vector operations (2D & 3D) Use this tool to explore vector algebra: magnitudes, dot product, angle between vectors, sum and difference, projection of one vector onto another, and (in 3D) the cross product and its magnitude (area). 2D vectors (x, y) 3D vectors (x, y, z) Vector A A x A y A z Empty fields are treated as 0. For 2D vectors, z is implicitly 0. Vector B B x B y B z You can model 2D vectors in 3D by setting z = 0. Compute vector operations Clear Load example Vector results summary – Dimension Vector A Vector B Magnitude |A| Magnitude |B| Dot product A · B Angle θ (degrees) Angle θ (radians) |A × B| (3D) Sum A + B Difference A − B Scalar projection of A on B Component of A along the direction of B: comp B (A) = (A · B)

Home » Math & Conversion » Coordinate Converters » Cartesian to Cylindrical Cartesian to Cylindrical Coordinate Converter Convert Cartesian coordinates (x, y, z) to Cylindrical coordinates (ρ, φ, z) instantly. Ideal for students, engineers, and professionals working with 3D coordinate systems. X Coordinate Y Coordinate Z Coordinate Results: ρ (rho): 0.00 φ (phi): 0.00° z: 0.00 Formulas \[ \rho = \sqrt{x^2 + y^2} \] \[ \phi = \arctan\left(\frac{y}{x}\right) \] \[ z = z \] How It Works The cylindrical coordinate system extends the two-dimensional polar coordinate system to three dimensions by adding a height (z) value. The conversion formulas transform Cartesian coordinates to their cylindrical equivalents. Example Calculation Given Cartesian coordinates (3, 4, 5): ρ = √(3² + 4²) = √(9 + 16) = √25 = 5 φ = arctan(4/3) ≈ 53.13° z = 5 Result: Cylindrical coordinates are (5, 53.13°, 5) Glossary of Terms ρ (rho): The radial distance from the z-axis (always non-negative) φ (phi): The azimuthal angle in the xy-plane from the positive x-axis z: The height above the xy-plane (same as Cartesian z) Cartesian coordinates: A coordinate system using (x, y, z) to specify positions in 3D space Cylindrical coordinates: A coordinate system using (ρ, φ, z) ideal for problems with cylindrical symmetry Frequently Asked Questions What is the cylindrical coordinate system? The cylindrical coordinate system extends 2D polar coordinates to 3D by adding a height (z) value. It's particularly useful for problems involving cylinders, spirals, and geometries with axial symmetry. Why use cylindrical coordinates? Cylindrical coordinates simplify calculations for problems with cylindrical symmetry, such as fluid flow in pipes, electromagnetic fields around wires, or motion along helical paths. How do I convert from cylindrical back to Cartesian coordinates? Use the reverse formulas: x = ρ cos(φ), y = ρ sin(φ), and z = z. Data Source and Methodology All calculations follow standard mathematical transformation formulas for coordinate system conversions, recognized in mathematics, physics, and engineering fields. Related Converters Cylindrical to Cartesian Cartesian to Polar Cartesian to Spherical Polar to Cartesian

Home » Construction & DIY » Project Layout & Design Construction & Project Layout Calculators Plan your DIY and construction projects with tools for room layout, floor plans, and space design. Room layout calculator Floor plan calculator Related categories Finance Calculators Health & Fitness Math & Conversions Lifestyle & Everyday

Home » Construction & DIY Calculators » Project Layout & Design Project Layout & Design Calculators Plan construction projects with layout calculators, design tools, and space planning utilities. Room layout calculator Calculate room layout calculator quickly and accurately. Floor plan calculator Calculate floor plan calculator quickly and accurately. Space planning calculator Calculate space planning calculator quickly and accurately. Furniture layout calculator Calculate furniture layout calculator quickly and accurately. Kitchen layout calculator Calculate kitchen layout calculator quickly and accurately. Bathroom layout calculator Calculate bathroom layout calculator quickly and accurately. Bedroom layout calculator Calculate bedroom layout calculator quickly and accurately. Living room layout calculator Calculate living room layout calculator quickly and accurately. Office layout calculator Calculate office layout calculator quickly and accurately. Garage layout calculator Calculate garage layout calculator quickly and accurately. Basement layout calculator Calculate basement layout calculator quickly and accurately. Attic layout calculator Calculate attic layout calculator quickly and accurately. Closet layout calculator Calculate closet layout calculator quickly and accurately. Pantry layout calculator Calculate pantry layout calculator quickly and accurately. Laundry room layout calculator Calculate laundry room layout calculator quickly and accurately. Mudroom layout calculator Calculate mudroom layout calculator quickly and accurately. Entryway layout calculator Calculate entryway layout calculator quickly and accurately. Hallway layout calculator Calculate hallway layout calculator quickly and accurately. Stairway layout calculator Calculate stairway layout calculator quickly and accurately. Porch layout calculator Calculate porch layout calculator quickly and accurately. Deck layout calculator Calculate deck layout calculator quickly and accurately. Patio layout calculator Calculate patio layout calculator quickly and accurately. Garden layout calculator Calculate garden layout calculator quickly and accurately. Landscape layout calculator Calculate landscape layout calculator quickly and accurately. Driveway layout calculator Calculate driveway layout calculator quickly and accurately. Walkway layout calculator Calculate walkway layout calculator quickly and accurately. Fence layout calculator Calculate fence layout calculator quickly and accurately. Gate layout calculator Calculate gate layout calculator quickly and accurately. Shed layout calculator Calculate shed layout calculator quickly and accurately. Greenhouse layout calculator Calculate greenhouse layout calculator quickly and accurately. Pool layout calculator Calculate pool layout calculator quickly and accurately. Spa layout calculator Calculate spa layout calculator quickly and accurately. Fire pit layout calculator Calculate fire pit layout calculator quickly and accurately. Outdoor kitchen layout calculator Calculate outdoor kitchen layout calculator quickly and accurately. Pergola layout calculator Calculate pergola layout calculator quickly and accurately. Gazebo layout calculator Calculate gazebo layout calculator quickly and accurately. Arbor layout calculator Calculate arbor layout calculator quickly and accurately. Trellis layout calculator Calculate trellis layout calculator quickly and accurately. Planter layout calculator Calculate planter layout calculator quickly and accurately. Raised bed layout calculator Calculate raised bed layout calculator quickly and accurately. Irrigation layout calculator Calculate irrigation layout calculator quickly and accurately. Drainage layout calculator Calculate drainage layout calculator quickly and accurately. Electrical layout calculator Calculate electrical layout calculator quickly and accurately. Plumbing layout calculator Calculate plumbing layout calculator quickly and accurately. HVAC layout calculator Calculate hvac layout calculator quickly and accurately. Lighting layout calculator Calculate lighting layout calculator quickly and accurately. Security layout calculator Calculate security layout calculator quickly and accurately. Network layout calculator Calculate network layout calculator quickly and accurately. Audio layout calculator Calculate audio layout calculator quickly and accurately. Theater layout calculator Calculate theater layout calculator quickly and accurately. Workshop layout calculator Calculate workshop layout calculator quickly and accurately. Studio layout calculator Calculate studio layout calculator quickly and accurately. Related categories Finance Calculators Health & Fitness Math & Conversions Lifestyle & Everyday

Home » Construction & DIY Calculators Construction & DIY Calculators Plan and execute construction and DIY projects with confidence. Calculate materials, estimate costs, plan layouts, and ensure your projects are built to last with our specialized construction calculators. Project Layout & Design Room layout calculator Calculate room layout calculator quickly and accurately. Floor plan calculator Calculate floor plan calculator quickly and accurately. Space planning calculator Calculate space planning calculator quickly and accurately. Furniture layout calculator Calculate furniture layout calculator quickly and accurately. Kitchen layout calculator Calculate kitchen layout calculator quickly and accurately. Bathroom layout calculator Calculate bathroom layout calculator quickly and accurately. View All Project Layout & Design Calculators Materials Estimation Concrete calculator Calculate concrete calculator quickly and accurately. Lumber calculator Calculate lumber calculator quickly and accurately. Paint calculator Calculate paint calculator quickly and accurately. Flooring calculator Calculate flooring calculator quickly and accurately. Tile calculator Calculate tile calculator quickly and accurately. Drywall calculator Calculate drywall calculator quickly and accurately. Epoxy resin calculator Accurately calculate the amount of epoxy resin needed for your DIY projects with our interactive calculator. Perfect for craftsmen and hobbyists. View All Materials Estimation Calculators Most popular Concrete Calculator Paint Calculator Flooring Calculator Lumber Calculator

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Home » Finance Calculators » Loans & Debt Loans & Debt Calculators Navigate the world of borrowing with calculators for personal loans, auto financing, student loans, and debt management strategies. Simple loan calculator Estimate payment size, total interest, and payoff timing for simple-interest loans. Loan amortization schedule calculator Calculate loan amortization schedule calculator quickly and accurately. Loan payoff calculator Calculate loan payoff calculator quickly and accurately. Debt-to-income (DTI) ratio calculator Calculate debt-to-income (dti) ratio calculator quickly and accurately. Personal loan calculator Calculate personal loan calculator quickly and accurately. APR calculator Calculate apr calculator quickly and accurately. Debt consolidation calculator Calculate debt consolidation calculator quickly and accurately. Student loan calculator Calculate student loan calculator quickly and accurately. Auto loan calculator Calculate auto loan calculator quickly and accurately. Car lease calculator Calculate car lease calculator quickly and accurately. Credit card payoff calculator Calculate credit card payoff calculator quickly and accurately. Debt snowball calculator Calculate debt snowball calculator quickly and accurately. Debt avalanche calculator Calculate debt avalanche calculator quickly and accurately. Loan comparison calculator Calculate loan comparison calculator quickly and accurately. Refinance calculator Calculate refinance calculator quickly and accurately. Balloon payment calculator Calculate balloon payment calculator quickly and accurately. Payday loan calculator Calculate payday loan calculator quickly and accurately. Title loan calculator Calculate title loan calculator quickly and accurately. Home equity loan calculator Calculate home equity loan calculator quickly and accurately. HELOC calculator Calculate heloc calculator quickly and accurately. Boat loan calculator Calculate boat loan calculator quickly and accurately. RV loan calculator Calculate rv loan calculator quickly and accurately. Motorcycle loan calculator Calculate motorcycle loan calculator quickly and accurately. Equipment loan calculator Calculate equipment loan calculator quickly and accurately. Construction loan calculator Calculate construction loan calculator quickly and accurately. Bridge loan calculator Calculate bridge loan calculator quickly and accurately. Hard money loan calculator Calculate hard money loan calculator quickly and accurately. Peer-to-peer loan calculator Calculate peer-to-peer loan calculator quickly and accurately. Installment loan calculator Calculate installment loan calculator quickly and accurately. Secured loan calculator Calculate secured loan calculator quickly and accurately. Unsecured loan calculator Calculate unsecured loan calculator quickly and accurately. Co-signer loan calculator Calculate co-signer loan calculator quickly and accurately. Related categories Finance Calculators Health & Fitness Math & Conversions Lifestyle & Everyday

Home » Finance Calculators » Mortgage & Real Estate Mortgage & Real Estate Calculators Make informed real estate decisions with mortgage payment calculators, affordability assessments, and refinancing analysis tools. Mortgage payment calculator Calculate mortgage payment quickly and accurately. Mortgage affordability calculator Calculate mortgage affordability quickly and accurately. 15 vs 30 year mortgage calculator Compare monthly payments, total interest, and payoff timelines between 15- and 30-year fixed mortgages. House affordability calculator Calculate house affordability quickly and accurately. Down payment calculator Calculate down payment quickly and accurately. Closing cost calculator Calculate closing cost quickly and accurately. PMI calculator Calculate pmi quickly and accurately. Rent vs buy calculator Calculate rent vs buy quickly and accurately. Property tax calculator Calculate property tax quickly and accurately. Home value calculator Calculate home value quickly and accurately. Home equity calculator Calculate home equity quickly and accurately. Cash-out refinance calculator Calculate cash-out refinance quickly and accurately. ARM calculator Calculate arm quickly and accurately. FHA loan calculator Calculate fha loan quickly and accurately. VA loan calculator Calculate va loan quickly and accurately. Jumbo loan calculator Calculate jumbo loan quickly and accurately. Investment property calculator Calculate investment property quickly and accurately. Related categories Finance Calculators Health & Fitness Math & Conversions Lifestyle & Everyday

Home » Finance Calculators » Retirement Retirement Calculators Plan contributions, pension income, SIPPs, IRAs, annuities, and long-term income goals with calculators built for modern retirement planning. Retirement calculator Project nest egg, sustainable withdrawals, Social Security, and contribution needs in today’s pounds. Pension payout calculator Translate pension savings into inflation-aware monthly income over your retirement horizon. Pension calculator (DB & DC) Estimate defined benefit income or project defined contribution balances with COLA and match options. SIPP calculator (UK) Model UK SIPP growth with contribution timing, annual increases, and inflation-adjusted projections. IRA calculator Track traditional IRA contributions, growth, catch-up limits, and retirement value. Roth IRA calculator Compare traditional vs. Roth IRA growth, contributions, and tax-free withdrawals. Annuity calculator Price immediate or deferred annuities, evaluate payouts, and compare present values. 401(k) calculator Model contributions, employer match, and retirement value for your 401(k). Roth conversion analyzer Compare taxable versus Roth conversions, break-even years, and projected balances. German pension calculator Estimate your monthly German statutory pension (Rente) with salary, age, and points. Need something else? Browse more retirement and tax planning tools or use the search bar above. Browse All Categories Retirement stats Calculators 8+ Category Finance Updated Weekly Related finance subcategories Investments 46 Taxes 108 Business Planning 64

Home » Finance Calculators Finance Calculators Master your financial decisions with our comprehensive suite of calculators. From mortgage payments and loan analysis to investment planning and retirement strategies, these tools help you make informed choices about your money. Loans & Debt Simple loan calculator Calculate simple loan calculator quickly and accurately. Loan amortization schedule calculator Calculate loan amortization schedule calculator quickly and accurately. Loan payoff calculator Calculate loan payoff calculator quickly and accurately. Debt-to-income (DTI) ratio calculator Calculate debt-to-income (dti) ratio calculator quickly and accurately. Personal loan calculator Calculate personal loan calculator quickly and accurately. APR calculator Calculate apr calculator quickly and accurately. Balloon loan calculator Calculate your balloon loan payments effortlessly. Ideal for those seeking clarity on personal loan repayments. 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Home » Lifestyle & Everyday Calculators » Time & Date Time & Date Calculators Master time calculations with date difference calculators, age calculators, and time zone converters. Age calculator Calculate age calculator quickly and accurately. Date calculator Calculate date calculator quickly and accurately. Time calculator Calculate time calculator quickly and accurately. Business days calculator Calculate business days calculator quickly and accurately. Workdays calculator Calculate workdays calculator quickly and accurately. Hours calculator Calculate hours calculator quickly and accurately. Minutes calculator Calculate minutes calculator quickly and accurately. Seconds calculator Calculate seconds calculator quickly and accurately. Time zone calculator Calculate time zone calculator quickly and accurately. Sunrise sunset calculator Calculate sunrise sunset calculator quickly and accurately. Moon phase calculator Calculate moon phase calculator quickly and accurately. Calendar calculator Calculate calendar calculator quickly and accurately. Leap year calculator Calculate leap year calculator quickly and accurately. Day of week calculator Calculate day of week calculator quickly and accurately. Week number calculator Calculate week number calculator quickly and accurately. Countdown calculator Calculate countdown calculator quickly and accurately. Timer calculator Calculate timer calculator quickly and accurately. Stopwatch calculator Calculate stopwatch calculator quickly and accurately. World clock calculator Calculate world clock calculator quickly and accurately. Time duration calculator Calculate time duration calculator quickly and accurately. Elapsed time calculator Calculate elapsed time calculator quickly and accurately. Time difference calculator Calculate time difference calculator quickly and accurately. Date difference calculator Calculate date difference calculator quickly and accurately. Date range calculator Calculate date range calculator quickly and accurately. Working hours calculator Calculate working hours calculator quickly and accurately. Overtime calculator Calculate overtime calculator quickly and accurately. Shift calculator Calculate shift calculator quickly and accurately. Schedule calculator Calculate schedule calculator quickly and accurately. Appointment calculator Calculate appointment calculator quickly and accurately. Meeting calculator Calculate meeting calculator quickly and accurately. Birthday calculator Calculate your age, next birthday, and more with our interactive Birthday Calculator. Related categories Finance Calculators Health & Fitness Math & Conversions Lifestyle & Everyday

Home » Lifestyle & Everyday Calculators Lifestyle & Everyday Calculators Simplify daily calculations with tools for time management, automotive needs, hobbies, and general life planning. From date calculations to hobby project planning, these calculators make everyday math effortless. Miscellaneous Tip calculator Calculate tip calculator quickly and accurately. Grade calculator Calculate grade calculator quickly and accurately. GPA calculator Calculate gpa calculator quickly and accurately. Discount calculator Calculate discount calculator quickly and accurately. Sales tax calculator Calculate sales tax calculator quickly and accurately. Markup calculator Calculate markup calculator quickly and accurately. Final grade calculator Calculate your final grade easily with our authoritative final grade calculator tool. Regular expression (regex) tester | engineering Test and validate regular expressions with our interactive Regex Tester. Perfect for software engineers and developers seeking a reliable tool. ROT13 cipher encoder/decoder Use our ROT13 calculator to encode or decode text using the ROT13 cipher, a simple letter substitution cipher used in cryptography. View All Miscellaneous Calculators Hobbies Knitting calculator Calculate knitting calculator quickly and accurately. Crochet calculator Calculate crochet calculator quickly and accurately. Sewing calculator Calculate sewing calculator quickly and accurately. Quilting calculator Calculate quilting calculator quickly and accurately. Embroidery calculator Calculate embroidery calculator quickly and accurately. Cross stitch calculator Calculate cross stitch calculator quickly and accurately. Candle wax calculator Calculate the perfect wax amount for your candles with our Candle Wax Calculator, an essential tool for DIY candle makers. Darts checkout calculator Use our interactive Darts Checkout Calculator to find the best checkout combinations and improve your darts game. Knitting gauge calculator Calculate your knitting gauge with precision using our interactive calculator designed for knitters. Poker odds calculator Calculate your winning odds in a poker game with our advanced Poker Odds Calculator. Enhance your strategy with accurate and reliable calculations. Sewing yardage calculator Calculate the exact amount of fabric you need for your sewing projects with our precise Sewing Yardage Calculator. Yarn calculator for knitting/crochet Calculate the amount of yarn needed for your knitting and crochet projects with our precise and easy-to-use Yarn Calculator. View All Hobbies Calculators Time & Date Age calculator Calculate age calculator quickly and accurately. Date calculator Calculate date calculator quickly and accurately. Time calculator Calculate time calculator quickly and accurately. Business days calculator Calculate business days calculator quickly and accurately. Workdays calculator Calculate workdays calculator quickly and accurately. Hours calculator Calculate hours calculator quickly and accurately. Birthday calculator Calculate your age, next birthday, and more with our interactive Birthday Calculator. View All Time & Date Calculators Automotive Gas mileage calculator Calculate gas mileage calculator quickly and accurately. Fuel economy calculator Calculate fuel economy calculator quickly and accurately. Trip cost calculator Calculate trip cost calculator quickly and accurately. Car payment calculator Calculate car payment calculator quickly and accurately. Auto insurance calculator Calculate auto insurance calculator quickly and accurately. Tire pressure calculator Calculate tire pressure calculator quickly and accurately. Car depreciation calculator Calculate how much your car will depreciate over time with our authoritative Car Depreciation Calculator. Fuel cost calculator | everyday life Calculate your fuel costs efficiently with our accurate and accessible Fuel Cost Calculator. Gas mileage (MPG) calculator Calculate your vehicle Oil change interval calculator Calculate the optimal interval for changing your car Tire size calculator Calculate the dimensions of your tire with our authoritative Tire Size Calculator. Determine the best fit for your vehicle with precision. View All Automotive Calculators Most popular Age Calculator Date Calculator Time Calculator Countdown Calculator

Home » Math & Conversion Calculators Math & Conversion Calculators Solve mathematical problems and convert between units with ease. From basic algebra to complex geometry, plus comprehensive unit conversions for length, weight, temperature, and more. Core Math & Algebra Percentage calculator Calculate percentage calculator quickly and accurately. Ratio calculator Calculate ratio calculator quickly and accurately. Proportion calculator Calculate proportion calculator quickly and accurately. Scientific calculator Calculate scientific calculator quickly and accurately. Fraction calculator Calculate fraction calculator quickly and accurately. Decimal calculator Calculate decimal calculator quickly and accurately. Derivative calculator Calculate derivatives easily with our interactive derivative calculator. Designed for students and professionals in calculus. Integral calculator Integral Calculator: Solve integrals effortlessly with our interactive tool. Perfect for students, educators, and professionals in calculus. Limit calculator An authoritative and interactive tool to calculate mathematical limits with precision and ease. Probability calculator | statistics & probability tool Professional probability calculator for simple events, independent events, Quartile calculator Calculate the first, second, and third quartiles of a dataset using our intuitive and accessible Quartile Calculator. View All Core Math & Algebra Calculators Geometry Area calculator Calculate area calculator quickly and accurately. Perimeter calculator Calculate perimeter calculator quickly and accurately. Volume calculator Calculate volume calculator quickly and accurately. Surface area calculator Calculate surface area calculator quickly and accurately. Circle calculator Calculate circle calculator quickly and accurately. Triangle calculator Calculate triangle calculator quickly and accurately. Cube calculator Explore this calculator to run scenarios and understand the underlying formulas. Equilateral triangle calculator Explore this calculator to run scenarios and understand the underlying formulas. Frustum of a cone volume calculator Calculate the volume of a frustum of a cone with ease using our interactive online calculator. Perfect for students, engineers, and geometry enthusiasts. Golden ratio calculator Calculate the golden ratio with precision using our interactive calculator. Perfect for designers, architects, and mathematicians. Inscribed angle calculator Explore this calculator to run scenarios and understand the underlying formulas. Isosceles triangle calculator Explore this calculator to run scenarios and understand the underlying formulas. Kite calculator Explore this calculator to run scenarios and understand the underlying formulas. Nonagon calculator Explore this calculator to run scenarios and understand the underlying formulas. Rectangular prism (cuboid) volume calculator Calculate the volume of a rectangular prism (cuboid) with our precise, accessible online tool. Perfect for students, educators, and professionals in geometry. Regular polygon area calculator Calculate the area of a regular polygon with our precise and easy-to-use tool. Suitable for students, teachers, and geometry enthusiasts. Rhombus calculator Explore this calculator to run scenarios and understand the underlying formulas. Scalene triangle calculator Explore this calculator to run scenarios and understand the underlying formulas. View All Geometry Calculators Measurement Unit Conversions Universal unit converter Convert any unit of measure across categories in one tool. Speed unit converter mph, kph, m/s, knots—convert between common speed units. Volume unit converter Gallons, liters, cubic feet/meters, and more. Weight/mass unit converter lbs, kg, stone, tons, and more mass units. Pressure unit converter Pascals, bar, atm, psi, torr—quick conversions. Area unit converter Acres, hectares, square feet/meters, and more. View All Measurement Unit Conversions Calculators Most popular Percentage Calculator Fraction Calculator Scientific Calculator Square Root Calculator

Home » Math & Conversions » Algebra » Combination Combination Calculator Find the number of ways to choose $k$ items from a set of $n$. Our tool also calculates **Permutations** and handles both **Repetition** and **No Repetition** scenarios. Total Items (n): Items Chosen (k): Calculate Combinations & Permutations Combinations (No Repetition) C(n, k) - Permutations (No Repetition) P(n, k) - Combinations (With Repetition) - Permutations (With Repetition) - Combinations vs. Permutations: When Does Order Matter? The first decision in combinatorics is determining if you are dealing with a combination or a permutation. Decision Matrix: Scenario Order Matters? Repetition Allowed? Formula Type Selecting a podium finish (1st, 2nd, 3rd) Yes (Permutation) No Permutation P(n, k) Selecting 3 friends for a party No (Combination) No Combination C(n, k) Locker code (digits can repeat) Yes (Permutation) Yes nᵏ Choosing donuts (multiple of the same type) No (Combination) Yes C(n + k - 1, k) Key Formulas (No Repetition) Permutation: $$ P(n, k) = \frac{n!}{(n - k)!} $$ Combination: $$ C(n, k) = \frac{n!}{k!(n - k)!} $$ Frequently Asked Questions (FAQ) What is the formula for a combination without repetition? The formula is: $C(n, k) = \frac{n!}{k!(n - k)!}$. This calculates the number of ways to choose $k$ items from a set of $n$ items, where the order of selection does not matter. When is a result a Permutation? A result is a permutation when the **order of the items matters**. For example, forming the word 'CAT' is different from forming 'ACT', even though they use the same letters. If you are picking a password, a winning race order, or assigning roles, use permutations. Can combinations have repetition? Yes. Combinations with repetition are used when the items being chosen can be selected more than once (e.g., choosing 3 cookies from 5 available flavors). The formula for combinations with repetition is $C(n + k - 1, k)$. Related Calculators Permutation Calculator Probability Calculator Factorial Calculator Mode Calculator Roots Calculator Logarithm Calculator Exponent Calculator

Home » Math & Conversions » Algebra » Factoring Factoring Calculator (Quadratic Polynomials) Factor any quadratic equation in the form **Ax² + Bx + C = 0**. Our tool provides the fully factored form and the step-by-step solution using the grouping method. Quadratic Factoring ($Ax^2 + Bx + C$) A B C Factor Factored Form Factors of a Single Integer Enter an Integer: Find Factors All Factors (Divisors) - Prime Factorization - What is Factoring in Math? Factoring is one of the most essential skills in algebra. It is the process of breaking down a complex expression (like a polynomial) into a product of simpler terms. When you factor a polynomial, you are essentially finding the binomials that, when multiplied together, result in the original expression. Key Factoring Methods (Polynomials) Greatest Common Factor (GCF): The first step for any expression. You factor out the largest term that all parts of the polynomial share (e.g., $3x^2 + 6x = 3x(x+2)$). Factoring Trinomials ($Ax^2 + Bx + C$): The grouping method (sum-product) is the most common technique taught to students. You split the middle term ($Bx$) into two terms ($Mx + Nx$) such that $M \times N = A \times C$ and $M + N = B$. Difference of Squares: A special case where $a^2 - b^2 = (a - b)(a + b)$ (e.g., $x^2 - 9 = (x-3)(x+3)$). Our calculator primarily uses the grouping method to factor polynomials of the form $Ax^2 + Bx + C$. Factoring Numbers vs. Factoring Expressions While the goal is the same—to find the parts that multiply back to the original—the application is different: Factoring Numbers: Leads to the Prime Factorization (the set of prime numbers that multiply to the original number, e.g., $24 = 2^3 \cdot 3$). Factoring Polynomials: Leads to the roots (solutions) of the equation. If $(x+3)(x+2)=0$, the roots are $x=-3$ and $x=-2$. Frequently Asked Questions (FAQ) What is the easiest way to factor a quadratic equation? The easiest way is often the **grouping (sum-product) method** for trinomials ($A=1$). For any quadratic, the fastest way to find the factors (roots) is by using the **Quadratic Formula** ($x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$), which can then be used to write the factored form. When should I use the factoring calculator? You should use the calculator to: 1) Practice the grouping method and verify your M and N numbers. 2) Check the final factored form of a homework problem. 3) Find the prime factorization of very large numbers. Related Calculators Quadratic Formula Roots Calculator Discriminant Calculator Exponent Calculator GCF Calculator Decimal to Fraction Matrix Calculator Logarithm Calculator Sequence Calculator Mode Calculator

Home » Math & Conversions » Algebra » Fibonacci Calculator Fibonacci Calculator (Sequence & Retracement) A 2-in-1 tool. Use the **Sequence Calculator** for math or the **Retracement Calculator** for financial trading (stocks, forex). Fibonacci Sequence Calculator (Math) Find the n-th number in the Fibonacci sequence and see the full list. Find Fibonacci Number at position (n): Calculate F(n) Result - Golden Ratio (Φ) Approx. - Sequence up to F(n): Enter a number (0-150) and click Calculate. Fibonacci Retracement Calculator (Trading) Calculate support and resistance levels for a financial asset (stock, forex, crypto) based on its high and low price. 1. Select Trend Direction Uptrend (Low to High) Downtrend (High to Low) 2. Low Price 3. High Price Calculate Levels The Two Types of "Fibonacci Calculator" The term "Fibonacci" is used in two very different fields: mathematics and financial trading. This page provides tools for both. 1. The Fibonacci Sequence (Mathematics) The Fibonacci sequence is a set of numbers where each number is the sum of the two preceding ones, starting from 0 and 1. It is a fundamental concept in mathematics. Fₙ = Fₙ₋₁ + Fₙ₋₂ Sequence: 0, 1, 1, 2, 3, 5, 8, 13, 21, 34, ... The "Golden Ratio" (Phi, or $\Phi$) is a special number approximately equal to 1.618. As the Fibonacci sequence progresses, the ratio between one number and the previous one (e.g., 34/21) gets closer and closer to $\Phi$. 2. Fibonacci Retracement (Financial Trading) In finance, traders use Fibonacci ratios to identify potential support (pullback) and resistance (extension) levels. After a significant price move (an "uptrend" or "downtrend"), the price will often "retrace" (pull back) to one of these key levels before continuing in the original direction. The key ratios are derived from the Golden Ratio: 61.8% (0.618): The Golden Ratio ($\Phi$ - 1) 38.2% (0.382): 1 - 0.618 23.6% (0.236): 0.382 * 0.618 50% (0.500): A common-sense middle level. Frequently Asked Questions (FAQ) What is the 10th number in the Fibonacci sequence? Using the sequence F₀=0, F₁=1... the 10th number (F₁₀) is 55. Our calculator can find any number in the sequence up to F₁₅₀. How do you calculate Fibonacci retracement? First, identify a High and Low price for a trend. Then, find the difference (the "Range"). For a downtrend, you subtract (Range × Ratio) from the High price to find support. For an uptrend, you add (Range × Ratio) to the Low price to find resistance. Example (Downtrend): High=$150, Low=$100. Range = $50. The 38.2% retracement level is: $150 - ($50 * 0.382) = $150 - $19.10 = $130.90. Related Calculators function Matrix roots logarithm mode range distance formula axis of symmetry coordinate sequence

Home » Math & Conversions » Algebra » Mean Mean Calculator (Average, Median, Mode, Range) Find the arithmetic **mean (average)**, median, mode, range, and standard deviation of any set of numbers. Enter your data below and click 'Calculate' to see the full statistical breakdown. Enter Your Numbers: Calculate Statistics Mean (Average) - Median - Mode - Range - Variance ($\sigma^2$) - Standard Deviation ($\sigma$) - How to Calculate the Mean (Average) The **Mean**, also known as the arithmetic average, is the most common measure of central tendency. It is calculated using a simple formula: $$ \text{Mean} (\bar{x}) = \frac{\sum x}{n} $$ Where $\sum x$ is the sum of all values in the data set, and $n$ is the count of values. Media vs. Mediana: Quando Usare Quale? **Mean (Media):** Use when your data is symmetrically distributed and does not contain major outliers. It uses every value in the set. **Median (Mediana):** Use when your data is highly skewed by extreme outliers (e.g., in income or housing prices). The median is less affected by these extreme values, providing a better measure of the "typical" center. What is Standard Deviation? The **Standard Deviation** ($\sigma$) is the most widely used measure of data dispersion, or how "spread out" the numbers are from the mean. A low standard deviation means the numbers are tightly clustered around the mean; a high standard deviation means they are widely spread. The **Variance** ($\sigma^2$) is the standard deviation squared. Frequently Asked Questions (FAQ) What is the formula for the Mean (Average)? The formula is: **Mean = Sum of all values ($\sum x$)

Home » Math & Conversions » Algebra » Mode Calculator Mean, Median, Mode, and Range Calculator Find the mean, median, mode, and range of a set of numbers. Enter your data (separated by commas, spaces, or new lines) and click "Calculate" to see the full statistical breakdown and step-by-step solution. Enter Your Numbers: Calculate Mean (Average) - Median - Mode - Range - What are Mean, Median, and Mode? Mean, Median, and Mode are the three primary "measures of central tendency" in statistics. They each describe the "center" of a data set in a different way. Mean (Average) The **mean** is the most common form of average. It is calculated by adding all the numbers in a set and then dividing by the count of numbers in that set. Mean = (Sum of all numbers)

Home » Math & Conversions » Algebra » Sequence Calculator Sequence Calculator Analyze any number sequence to find the formula, next terms, and sum. Our tool automatically detects **Arithmetic**, **Geometric**, and **Quadratic** sequences. Enter Your Number Sequence: Enter at least 3 numbers, separated by commas or spaces. Analyze Sequence How to Find the Rule for a Sequence This calculator is a powerful tool that automatically checks for the most common types of mathematical sequences. Here are the formulas it uses: 1. Arithmetic Sequence A sequence is **arithmetic** if the difference between terms is constant. This is called the "common difference" (d). Example: 5, 8, 11, 14... (The common difference `d` is 3). Explicit Formula (nth term): `aₙ = a₁ + (n-1)d` Sum Formula (Series): `Sₙ = n/2 * (2a₁ + (n-1)d)` 2. Geometric Sequence A sequence is **geometric** if the ratio between terms is constant. This is called the "common ratio" (r). Example: 3, 6, 12, 24... (The common ratio `r` is 2). Explicit Formula (nth term): `aₙ = a₁ * rⁿ⁻¹` Sum Formula (Series): `Sₙ = a₁ * (1 - rⁿ)

Home » Math & Conversions » Algebra » Square Root Square Root and Nth Root Calculator Find the square root ($\sqrt{x}$) or any arbitrary N-th root ($\sqrt[N]{x}$) of a number. Click 'Calculate' to see the exact decimal result and the simplified radical form. N-th Root (N): Radicand (X): Calculate Square Root ($\sqrt{X}$) - Nth Root ($\sqrt[N]{X}$) - Understanding Square Roots and Nth Roots The square root ($\sqrt{x}$) is the most common radical function. The **Nth Root** ($\sqrt[N]{x}$) is the generalized function that includes the square root (where $N=2$) and the cube root (where $N=3$). The Nth Root Formula The calculation is essentially finding a number $R$ that, when multiplied by itself $N$ times, equals the original number $X$. $$ R = \sqrt[N]{X} \quad \text{which is equivalent to} \quad R^N = X $$ Since most calculators work with exponents, the practical formula is: $$ \sqrt[N]{X} = X^{\frac{1}{N}} $$ Simplifying Radicals (Square Roots) When solving algebra problems, you often need the **simplified radical form** (e.g., $5\sqrt{2}$) rather than the decimal answer (e.g., 7.071). To simplify a square root, you find the largest perfect square factor of the radicand. Example: Simplify $\sqrt{50}$ Identify factors of 50: (1, 50), (2, 25), (5, 10). Find the largest perfect square factor: 25. Rewrite the radical: $\sqrt{50} = \sqrt{25 \cdot 2}$. Take the square root of the perfect square: $\sqrt{25} \cdot \sqrt{2} = 5\sqrt{2}$. Our calculator performs this simplification for you. Frequently Asked Questions (FAQ) What is the formula for square root? The formula is $\sqrt{X}$. It is calculated by raising the number $X$ to the power of $\frac{1}{2}$ ($X^{1/2}$). Is $\sqrt{2}$ an irrational number? Yes. $\sqrt{2}$ (approximately 1.41421...) is an **irrational number**. This means it cannot be expressed as a simple fraction (a/b) and its decimal expansion is non-terminating and non-repeating. Can you find the square root of a negative number? In real numbers, **no**. The square root of a negative number (e.g., $\sqrt{-4}$) is an **imaginary number** ($2i$). Our calculator returns an error for negative inputs. Related Calculators Roots Calculator (Nth Root) Exponent Calculator Logarithm Calculator Function Calculator Matrix Calculator Mode Calculator Range Calculator Factoring Calculator Decimal to Fraction

Home » Math & Conversions » Geometry » Pentagon Calculator Regular Pentagon Calculator Calculate all properties of a regular pentagon. Enter one measurement (Side, Apothem, or Circumradius) to find the Area, Perimeter, and Angles. Input (Enter ONE Value) Side Length (s): Apothem (a): Circumradius (R): Calculate Properties Results Area (A) Perimeter (P) Apothem (a) Circumradius (R) Internal Angle 108° Formulas for a Regular Pentagon A regular pentagon is a polygon with five equal sides and five equal internal angles ($\alpha = 108^\circ$). All calculations are based on the length of the side ($s$), the apothem ($a$), or the circumradius ($R$). Area Calculation (A) The standard formula uses the side length ($s$): $$ A = \frac{1}{4} s^2 \sqrt{25 + 10\sqrt{5}} $$ Alternatively, using the apothem ($a$) and perimeter ($P$): $$ A = \frac{1}{2} P a \quad \text{where} \quad P = 5s $$ Key Relationships (Side, Apothem, Radius) The calculator finds the relationships using trigonometry ($\theta = 360^\circ

Home » Math & Conversions » Geometry » Trapezoid Calculator Trapezoid Calculator Calculate the area, perimeter, angles, and diagonals of any trapezoid (also known as a trapezium). Enter the lengths of the four sides and the height (if known). Input Parameters Base a (Top parallel side): Base b (Bottom parallel side): Side c (Left oblique side): Side d (Right oblique side): Height h (Perpendicular): Decimal Precision: Calculate Trapezoid Properties Results Area (A) Perimeter (P) Height (h) Trapezoid Type Angles and Diagonals Angle α (Base a, Left) Angle β (Base b, Left) Angle γ (Base b, Right) Angle δ (Base a, Right) Trapezoid Formulas and Definitions A trapezoid (or trapezium) is a quadrilateral with at least one pair of parallel sides. These parallel sides are called the bases ($a$ and $b$). Area and Perimeter Formulas Area (A): The area is the average of the two parallel bases multiplied by the perpendicular height. $$ A = \frac{a + b}{2} \cdot h $$ Perimeter (P): The perimeter is the sum of all four side lengths. $$ P = a + b + c + d $$ Types of Trapezoids Isosceles Trapezoid: The non-parallel sides ($c$ and $d$) are equal in length ($c=d$). The base angles are also equal ($\alpha = \delta$ and $\beta = \gamma$). Right Trapezoid: Has at least two right angles (90°). This occurs when one of the non-parallel sides ($c$ or $d$) is perpendicular to the parallel bases. Scalene Trapezoid: All four sides are of different lengths. Frequently Asked Questions (FAQ) What is the formula for the area of a trapezoid? The area formula is: $A = \\frac{a + b}{2} \\cdot h$, where $a$ and $b$ are the parallel bases and $h$ is the height. How do you find the height of a trapezoid? If the height ($h$) is not given, it can be calculated using trigonometry or the Pythagorean theorem after splitting the trapezoid into a rectangle and one or two right triangles. This requires knowing the length of all four sides. What is the sum of the angles in a trapezoid? Like all quadrilaterals, the sum of the interior angles in any trapezoid is always **360 degrees**. Related Calculators Solid Geometry Calculator (All Shapes) ellipse Triangle cylinder Chord Length angle central angle Midpoint pentagon Cone Volume

Home » Math & Conversions » Geometry » Triangle Calculator Universal Triangle Solver (SSS, SAS, ASA, AAS) Enter any three parameters of the triangle (three sides, two sides and an angle, or two angles and a side) to find all missing values, the area, the perimeter, and the type of triangle. Input (Enter 3 Values) Side a: Side b: Side c: Angle α (degrees): Angle β (degrees): Angle γ (degrees): Solve Triangle Triangle Solutions Side a Side b Side c Angle α Angle β Angle γ Area Perimeter Type Triangle Solving: Law of Sines vs. Law of Cosines To solve a triangle (find all missing sides and angles), you need at least three pieces of information, including at least one side. The core of triangle solving relies on two trigonometric laws. Law of Cosines (for SSS and SAS Cases) The Law of Cosines is typically used first when you know all three sides (SSS) or two sides and the angle *between* them (SAS). $$ a^2 = b^2 + c^2 - 2bc \cos(\alpha) $$ Law of Sines (for ASA, AAS, and to finish solutions) The Law of Sines is used when you know a side and its opposite angle, or two angles and a side (ASA or AAS). $$ \frac{a}{\sin(\alpha)} = \frac{b}{\sin(\beta)} = \frac{c}{\sin(\gamma)} $$ The **Sum of Angles** is always $180^\circ$: $\alpha + \beta + \gamma = 180^\circ$. Area Calculation The calculator uses the most efficient area formula based on your inputs: **SAS Case:** Area = $\frac{1}{2}ab\sin(\gamma)$ **SSS Case:** Area is calculated using Heron's Formula (requires all three sides). Frequently Asked Questions (FAQ) What are the three ways to classify a triangle? Triangles are classified in two ways: **By Sides:** Equilateral (3 equal sides), Isosceles (2 equal sides), Scalene (no equal sides). **By Angles:** Acute (all angles < 90°), Obtuse (one angle > 90°), Right (one angle = 90°). Can a triangle have two right angles? No. Since the sum of angles in any triangle must be $180^\circ$, having two right angles ($90^\circ + 90^\circ = 180^\circ$) would leave $0^\circ$ for the third angle, making it impossible to form a closed shape. Related Calculators Right Triangle Calculator Law of Sines Calculator Law of Cosines Calculator Pythagorean Theorem Calculator Heron's Formula Calculator Trapezoid Calculator Area Calculator Perimeter Calculator

Home » Math & Conversions » Measurement Unit Conversions 122 calculators available Specialized tools for Measurement Unit Conversions. Search above to find specific calculators. Speed Unit Converter (mph, kph, m/s, knots) Volume Unit Converter (Gallons, Liters, cubic ft, cubic m) Universal Unit Converter Weight/Mass Unit Converter (lbs, kg, stone, tons) Cubic meters to liters converter Tons to pounds converter Pints to liters converter Force & torque Biblical units converter Beaufort wind scale converter Gauss to tesla converter Millibars to psi converter Pressure unit converter (pascal, bar, atm, psi) Magnetism Area unit converter (acres, hectares, sq ft, sq m) Points to pixels converter Leagues to miles/kilometers converter Micrograms to milligrams converter Fluid ounces to milliliters converter Torr to pascals converter Energy & heat Slug to kilogram converter Psi to pascals converter Cubic feet per minute (cfm) to liters per second converter Troy ounces to grams converter Awg to mm/mm² converter Luggage weight converter Bushels to liters converter Food-specific gravity to weight converter Poise to pascal-second converter Color code converter (hex, rgb, hsl, cmyk) Curies to becquerels converter Light-years to miles converter Grams to cups Bandwidth & data rate Newton-meters to foot-pounds converter Historical & specialized units Barns to square meters converter Moles to grams Inches of mercury to psi converter Ems to pixels converter Speed Rems to sieverts converter Weight & mass converters Megabytes to gigabytes converter Ancient greek units converter Cooking measurement converter Viscosity unit converter (poise, stokes) Calorie to btu converter Square yards to square feet converter Teaspoons to milliliters converter Bits to bytes converter Cubic feet to gallons converter Angle unit converter (degrees, radians, gradians) Mass & weight Tablespoons to milliliters converter Pixels to ems converter Btu to calories converter Therms to btu converter Volume & capacity Quarts to cups converter Slugs to kilograms converter Inches to picas converter Electronvolts to joules converter Length unit converter (miles, km, feet, meters) Pints to cups converter Power unit converter (watts, horsepower, btu/hr) Rankine to celsius converter Pressure Lumens to watts converter Kilobits per second to megabits per second converter Feet of water to psi converter Density Ergs to joules converter Chains to feet converter Fathoms to feet converter Steps to miles Foot-pounds to newton-meters converter Square miles to square kilometers converter Cords to cubic feet converter Hogsheads to gallons converter Rods to feet converter Grams to moles Nanometers to meters converter Gallons per minute (gpm) to liters per second converter Kilobytes to megabytes converter Flow rate Roods to acres converter Feet and inches to centimeters converter Pounds to stone converter Temperature Grains to grams converter Mohs hardness scale converter Data transfer rate converter Townships to square miles converter Data storage Square feet to acres converter Pennyweight to grams converter Gigabytes to terabytes converter Scruples to grams converter Torque unit converter (n-m, ft-lb, in-lb) Typography & web Viscosity Pascals to psi converter Area Kilopascals to psi converter Data storage converter (bits, bytes, kb, mb, gb, tb, pb) Square inches to square feet converter Furlongs to miles converter Ounces to tablespoons converter Hectares to acres converter Quarts to liters converter Radiation & dose Petabytes to terabytes converter Micrometers to millimeters converter Cubic meter (cbm) Dram to grams converter Soil salinity (ec) conversion Energy unit converter (joules, calories, kwh, btu) Centipoise to pascal-seconds converter Ancient roman units converter Rems to pixels converter Can't find what you're looking for? We're constantly adding new calculators. Browse our other categories or use the search function above. Browse All Categories Measurement unit conversions stats Calculators 122 Category Math & Conversions Updated Daily Related in math & conversions All Math & Conversions Geometry

Home » Math & Conversions » Unit Conversions » Acres to Square Feet (ac to ft²) Converter Acres to Square Feet (ac to ft²) Converter A complete land area converter for real estate and planning. Use the 'Area from Dimensions' tool for a plot, or the multi-unit converter for `acres`, `ft²`, `m²`, `hectares`, and `yd²`. Calculate Area from Dimensions Enter plot dimensions in feet to instantly find the area in both ft² and acres. Length (feet) Width (feet) Area (Square Feet) Area (Acres) Multi-Unit Area Converter Enter a value in any field to convert between all common land units. Acres Square Feet (ft²) Square Meters (m²) Hectares (ha) Square Yards (yd²) How to Convert Acres to Square Feet The conversion between acres and square feet is a fixed definition used in the United States and other countries using imperial or US customary units. It is essential for real estate, land surveying, and agriculture. The Exact Formula The relationship is an exact integer, defined by law: 1 Acre = 43,560 Square Feet (ft²) To convert from acres to square feet, you **multiply the acre value by 43,560**. Formula: ft² = acres × 43,560 Example: 0.25 acres × 43,560 = 10,890 ft² How to Convert Square Feet to Acres To convert the other way, you **divide the square feet value by 43,560**. Acres = Square Feet

Home » Math & Conversions » Unit Conversions » Angstroms to Nanometers (Å to nm) Converter Angstroms to Nanometers (Å to nm) Converter A simple, bidirectional converter for angstroms and nanometers. Below, find an advanced wavelength converter for scientific applications (Å, nm, µm, and eV). Simple Å ↔ nm Conversion Angstroms (Å) Nanometers (nm) Advanced Wavelength & Energy Converter For scientific use. Enter a value in any field to convert between angstroms, nanometers, micrometers, and electronvolts (eV). The tool will also identify the corresponding part of the electromagnetic spectrum. Angstroms (Å) Nanometers (nm) Micrometers (µm) Electronvolts (eV) Electromagnetic Spectrum - How to Convert Angstroms to Nanometers The conversion between angstroms (Å) and nanometers (nm) is one of the simplest in physics, as they are directly related by a factor of 10. The Conversion Formula The relationship is fixed and exact: 1 Nanometer (nm) = 10 Angstroms (Å) Therefore, to convert from angstroms to nanometers, you **divide the angstrom value by 10**. Formula: nm = Å

Home » Math & Conversions » Unit Conversions » Atmospheres to PSI Converter Atmospheres to PSI (atm to psi) Converter A complete pressure unit conversion tool. Enter a value in any field to instantly convert between **Atmospheres (atm)**, **PSI**, **Bar**, **Kilopascals (kPa)**, and **Torr (mmHg)**. Atmospheres (atm) Pounds per Square Inch (PSI) Bar Kilopascals (kPa) Torr (mmHg) Value in Pascals (Pa) - The Difference: atm vs. bar vs. Pascal The main source of confusion when converting pressure is the similarity between "atm" and "bar". They are close in value but are not the same. Pascal (Pa): This is the base SI unit for pressure. It's very small (1 N/m²), so we almost always use **Kilopascals (kPa)** (1,000 Pa) or **Megapascals (MPa)** (1,000,000 Pa). Bar: A simple metric unit. 1 bar = 100,000 Pa exactly. This is very close to atmospheric pressure. Standard Atmosphere (atm): This is an older unit, defined as the *average* atmospheric pressure at sea level. 1 atm = 101,325 Pa exactly (or 1.01325 bar). PSI (Pounds per Square Inch): The Imperial unit, common in the US, defined by force (pounds) over area (square inches). Torr (mmHg): A unit based on the pressure exerted by one millimeter of mercury, commonly used in meteorology and for vacuum pressures. 1 atm = 760 Torr . Conversion Formulas All conversions are based on the Pascal (Pa): 1 atm = 101,325 Pa 1 PSI = 6,894.757 Pa 1 bar = 100,000 Pa 1 kPa = 1,000 Pa 1 Torr = 133.322 Pa From this, we can find the direct formula: PSI = atm × 14.6959... atm = PSI × 0.068046... Common Pressure Conversion Table Measurement atm PSI kPa Bar Absolute Vacuum 0 0 0 0 Standard Atmosphere 1 atm 14.696 psi 101.325 kPa 1.013 bar Car Tire Pressure ~2.2 atm ~32 psi ~220 kPa ~2.2 bar Home Water Pressure ~3.4 atm ~50 psi ~345 kPa ~3.4 bar Frequently Asked Questions (FAQ) What is the formula for atm to PSI? The exact formula is: PSI = atm × 14.6959487755 . For most calculations, multiplying by 14.696 is sufficient. How many PSI are in 1 atm? There are **14.696 PSI** (Pounds per Square Inch) in 1 standard atmosphere (atm). Is 1 bar the same as 1 atm? No. They are very close, but 1 standard atmosphere (atm) is slightly larger than 1 bar. 1 atm = 1.01325 bar . Related Calculators Millimeters to Inches Converter Typography Unit Converter (px, pt, em, rem, %) Kilograms to Pounds Converter Centimeters to Inches Converter BTU to kWh Converter Angstroms to Nanometers Converter Kelvin to Fahrenheit Converter Degrees to Radians Converter Cups to Tablespoons Converter Barrels to Gallons Converter

Home » Math & Conversions » Unit Conversions » Bar to PSI Converter Bar to PSI Converter A complete pressure unit conversion tool for tire pressure and engineering. Enter a value in any field to instantly convert between **Bar**, **PSI**, **Kilopascals (kPa)**, and **Atmospheres (atm)**. Bar Pounds per Square Inch (PSI) Kilopascals (kPa) Atmospheres (atm) How to Convert Bar to PSI Converting from bar to Pounds per Square Inch (PSI) is the most common pressure conversion for drivers, as European and US tire pressure gauges use different units. The conversion is straightforward. The Conversion Formula The relationship between bar and PSI is based on their definitions in Pascals: 1 bar = 100,000 Pascals (Pa) 1 PSI = 6,894.757 Pascals (Pa) From this, we get the direct conversion factor: Bar to PSI: PSI = bar × 14.50377 PSI to Bar: bar = PSI

Home » Math & Conversions » Unit Conversions » Barrels to Gallons Converter Barrels to Gallons Converter The term "barrel" is ambiguous. Select the type of barrel you want to convert to instantly see the result in US gallons. Select Barrel Type: US Oil Barrel (bbl) US Liquid Barrel US Beer Barrel US Dry Barrel UK (Imperial) Barrel Barrels US Gallons (Liquid) Which "Barrel" Are You Converting? Unlike units like "meter" or "kilogram," the "barrel" is not a single defined unit. Its volume depends entirely on the context. Our calculator defaults to the **US Oil Barrel**, which is the most common international standard, but you may be referring to one of several others. Conversion Formulas Here are the exact conversion factors used by this calculator: 1 US Oil Barrel (bbl) = 42 US Gallons This is the standard for measuring crude oil and other petroleum products. 1 US Liquid Barrel = 31.5 US Gallons This is a less common standard, sometimes used for other liquids like chemicals or food products. 1 US Beer Barrel = 31 US Gallons This unit is legally defined in the US for tax purposes related to beer. 1 US Dry Barrel = 105 dry quarts (28.378 US Gallons) Used for some dry goods. 1 UK (Imperial) Barrel = 36 Imperial Gallons (43.234 US Gallons) Used in the United Kingdom. Note that an Imperial gallon is larger than a US gallon. Conversion Table (Oil Barrels) This table shows the conversion for the most common unit, the US Oil Barrel (bbl). US Oil Barrels (bbl) US Gallons 1 bbl 42 gal 2 bbl 84 gal 5 bbl 210 gal 10 bbl 420 gal 50 bbl 2,100 gal 100 bbl 4,200 gal Frequently Asked Questions (FAQ) How many gallons are in a barrel of oil? There are exactly **42 US gallons** in one standard barrel of oil (bbl). How many gallons are in a barrel of beer? In the United States, a standard barrel of beer is **31 US gallons**. This is different from a "keg," which is typically a half-barrel (15.5 gallons). Why is an oil barrel 42 gallons? The 42-gallon standard dates back to the 1870s in Pennsylvania. Early oil producers used 42-gallon wooden barrels (originally used for whiskey) for transport. This size became the standard for the industry to avoid disputes and ensure consistent measurement for trade. Related Calculators Millimeters to Inches Converter Typography Unit Converter (px, pt, em, rem, %) Kilograms to Pounds Converter Centimeters to Inches Converter BTU to kWh Converter Angstroms to Nanometers Converter Kelvin to Fahrenheit Converter Degrees to Radians Converter Cups to Tablespoons Converter Meters to Nanometers Converter

Home » Math & Conversions » Unit Conversions » BTU to Joules (J) Converter BTU to Joules (J) Converter A dual-purpose scientific converter for **Energy (BTU, J, kJ)** and **Power (BTU/hr, W, kW)**. Enter a value in any field to convert instantly. Energy Conversion (BTU ↔ J ↔ kJ) Use this to convert total energy (e.g., thermal energy, battery capacity). BTU (British Thermal Units) Joules (J) Kilojoules (kJ) Power Conversion (BTU/hr ↔ W ↔ kW) Use this for appliance ratings (e.g., heaters, AC units). Note: **1 Watt = 1 Joule per second (J/s)**. BTU per Hour (BTU/hr) Watts (W) Kilowatts (kW) BTU vs. BTU/hr: The Critical Difference (Energy vs. Power) The most common error in this conversion is confusing **Energy** (measured in BTU or Joules) with **Power** (measured in BTU/hr or Watts). They are not the same. Energy (BTU, Joule): This is a *total amount* of energy. Think of it like the amount of heat stored in a hot object. Power (BTU/hr, Watt): This is the *rate* at which energy is used. A 100-Watt light bulb uses 100 Joules of energy *every second*. (1 W = 1 J/s). This page provides both calculators. If you are converting a static energy value, use the first calculator. If you are converting an appliance rating (like a heater or AC unit), use the second calculator. Conversion Formulas The conversion factors are precise: Energy: 1 BTU = 1055.056 Joules (J) Power: 1 BTU/hr = 0.293071 Watts (W, or J/s) To convert from BTU to Joules, you **multiply by 1055.056**. To convert from BTU/hr to Watts, you **multiply by 0.293071**. Common Conversion Chart BTU (Energy) Joules (J) BTU/hr (Power) Watts (W) 1 BTU 1,055 J 1 BTU/hr 0.293 W 100 BTU 105,506 J 1,000 BTU/hr 293 W 1,000 BTU 1.055 MJ (MegaJ) 3,412 BTU/hr 1,000 W (1 kW) 10,000 BTU 10.55 MJ 10,000 BTU/hr 2,931 W (2.93 kW) Frequently Asked Questions (FAQ) What is the formula for BTU to Joules (J)? The formula is: Joules = BTU × 1055.056 . How many Joules are in 1 BTU? There are **1,055.056 Joules** (or 1.055 kilojoules) in 1 British Thermal Unit (BTU). What is the difference between BTU/hr and Joules/second (Watts)? BTU/hr measures **Power** (the rate of energy use), just as a Watt (W) does. A Watt is defined as 1 Joule per second. The conversion is: 1 BTU/hr = 0.293071 Watts (J/s). Related Calculators BTU to kWh Converter kWh to BTU Converter Calories to Joules Converter PSI to Bar Converter Kelvin to Fahrenheit Converter Fahrenheit to Kelvin Converter Angstroms to Nanometers Converter Nanometers to Meters Converter Millimeters to Inches Converter Kilograms to Pounds Converter

Home » Math & Conversions » Unit Conversions » BTU to kWh Converter BTU to kWh Converter Convert energy from British Thermal Units (BTU) to kilowatt-hours (kWh) or calculate the energy cost of an appliance like an air conditioner. Energy Conversion (BTU ↔ kWh) Use this for converting pure energy units. This is **not** for appliance ratings (see next calculator). BTU Kilowatt-Hours (kWh) AC

Home » Math & Conversions » Unit Conversions » Calories to Joules (cal to J) Converter Calories to Joules (cal to J) Converter A 3-way energy converter. Enter a value in any field to instantly convert between **Joules (J)**, **Calories (cal)**, and **Kilocalories (kcal)** (food Calories). Joules (J) Calories (cal) (small/gram) Kilocalories (kcal) (Food Calories/Cal) The Difference: Calorie (cal) vs. Calorie (kcal) The main source of confusion in this conversion is that the word "calorie" is used for two different units of energy. This tool converts both, eliminating the confusion. Calorie (cal): This is the "small calorie" or "gram calorie" used in physics and chemistry. It's the energy needed to raise the temperature of 1 gram of water by 1°C. Kilocalorie (kcal): This is the "large calorie" or "food Calorie" (with a capital C). It's the unit you see on nutrition labels. 1 kcal = 1,000 cal . Conversion Formulas The conversion is based on the international standard for the thermochemical calorie: 1 Calorie (cal) = 4.184 Joules (J) From this, all other conversions are derived: Joules to Calories: cal = J

Home » Math & Conversions » Unit Conversions » Celsius to Fahrenheit (°C to °F) Converter Celsius to Fahrenheit (°C to °F) Converter A complete temperature conversion tool. Enter a value in any field to instantly convert between **Celsius (°C)**, **Fahrenheit (°F)**, and **Kelvin (K)**. Celsius (°C) Fahrenheit (°F) Kelvin (K) Calculation Steps: Enter a value to see the step-by-step conversion. How to Convert Celsius to Fahrenheit Converting from Celsius (°C) to Fahrenheit (°F) is essential for anyone used to metric temperatures traveling in the US. The conversion can be done with an exact formula or a quick mental math trick for weather. The Exact Formula To convert Celsius to Fahrenheit, you first multiply the Celsius temperature by 9/5 (or 1.8), then add 32. °F = (°C × 9/5) + 32 Example: Convert 25°C (room temperature) to °F. 1. 25 × (9/5) = 45 2. 45 + 32 = 77 °F The Easy Mental Math Trick For a quick estimate (great for weather), you can **"double it and add 30"**. °F ≈ (°C × 2) + 30 Example: Convert 25°C to °F using the trick. Double it: 25 × 2 = 50 Add 30: 50 + 30 = 80 °F This provides a very fast and close estimate (80°F) to the real answer (77°F). Key Temperature Points Point Celsius (°C) Fahrenheit (°F) Kelvin (K) Water Boils 100 °C 212 °F 373.15 K Hot Day 30 °C 86 °F 303.15 K Water Freezes 0 °C 32 °F 273.15 K Absolute Zero -273.15 °C -459.67 °F 0 K Frequently Asked Questions (FAQ) What is the formula for Celsius (°C) to Fahrenheit (°F)? The formula is: Fahrenheit = (Celsius × 9/5) + 32 . You can also use 1.8 instead of 9/5. What is the quick way to convert °C to °F? The fastest mental math trick is to **double the Celsius temperature and add 30**. (e.g., 10°C * 2 = 20, + 30 = 50°F). The exact answer is 50°F, so this trick is very accurate for common temperatures. What is 0 degrees Celsius in Fahrenheit? 0°C is exactly **32°F**, which is the freezing point of water. Related Calculators Kelvin to Fahrenheit Converter Fahrenheit to Kelvin Converter Kelvin to Celsius Converter Fahrenheit to Celsius Converter BTU to kWh Converter Millimeters to Inches Converter Kilograms to Pounds Converter Centimeters to Inches Converter

Home » Math & Conversions » Unit Conversions » Celsius to Kelvin (°C to K) Converter Celsius to Kelvin (°C to K) Converter A complete temperature conversion tool. Enter a value in any field to instantly convert between **Celsius (°C)**, **Kelvin (K)**, and **Fahrenheit (°F)**. Celsius (°C) Kelvin (K) Fahrenheit (°F) Calculation Steps: Enter a value to see the step-by-step conversion. How to Convert Celsius to Kelvin The conversion from Celsius (°C) to Kelvin (K) is the simplest temperature conversion because the *size* of one degree Celsius is identical to the size of one Kelvin. The only difference is their "zero" point. The Formula (Celsius to Kelvin) To convert from Celsius to Kelvin, you simply add 273.15. K = °C + 273.15 Example: Convert 100°C (the boiling point of water) to Kelvin. 100 °C + 273.15 = 373.15 K The Formula (Kelvin to Celsius) To convert from Kelvin to Celsius, you subtract 273.15. °C = K - 273.15 What is Absolute Zero? The Kelvin scale is an "absolute" scale, meaning its zero point (0 K) is **Absolute Zero**. This is the theoretical temperature at which all atomic and molecular motion ceases. There is no temperature colder than 0 K, which is why the Kelvin scale has no negative numbers. Celsius and Fahrenheit are "relative" scales, as their zero points (0°C and 0°F) were set for convenience (e.g., the freezing point of water). Key Temperature Points Point Celsius (°C) Kelvin (K) Fahrenheit (°F) Water Boils 100 °C 373.15 K 212 °F Human Body (Avg) 37 °C 310.15 K 98.6 °F Room Temperature 20-25 °C ~293-298 K 68-77 °F Water Freezes 0 °C 273.15 K 32 °F Absolute Zero -273.15 °C 0 K -459.67 °F Frequently Asked Questions (FAQ) What is the formula for Celsius (°C) to Kelvin (K)? The formula is: Kelvin = Celsius + 273.15 . (K = °C + 273.15). What is 0 Celsius in Kelvin? 0 °C (the freezing point of water) is equal to 273.15 K . What is 0 Kelvin in Celsius? 0 Kelvin (K) is equal to -273.15 °C . This is known as Absolute Zero. Related Calculators Kelvin to Fahrenheit Converter Fahrenheit to Kelvin Converter Celsius to Fahrenheit Converter Fahrenheit to Celsius Converter BTU to kWh Converter Millimeters to Inches Converter Kilograms to Pounds Converter Centimeters to Inches Converter

Home » Math & Conversions » Unit Conversions » Centimeters to Inches (cm to in) Converter Centimeters to Inches (cm to in) Converter A fast, bidirectional converter for centimeters, inches, and feet. Enter a value in any of the first three fields to see all values update instantly. Centimeters (cm) = Inches (decimal) = Feet (ft) + Inches (in) ft in Result as Feet + Fractional Inches (approx.) - How to Convert Centimeters to Inches Converting from centimeters (cm) to inches (in) is simple once you know the exact conversion factor. For height, it's also useful to convert the total inches into feet and inches. The Exact Formula: CM to Inches The international standard for the inch is exactly **2.54 centimeters**. 1 inch = 2.54 cm To convert from cm to inches, you divide your centimeter value by 2.54. Alternatively, you can multiply by the reciprocal (1

Home » Math & Conversions » Unit Conversions » Cups to Tablespoons (tbsp) Converter Cups to Tablespoons (tbsp) & Teaspoons (tsp) Converter Your all-in-one kitchen conversion tool. Enter a value in any field to instantly convert between cups, tablespoons (tbsp), and teaspoons (tsp). Cups (US Customary) = Tablespoons (tbsp) = Teaspoons (tsp) How to Convert Cups to Tablespoons For US customary cooking measurements, the conversion is very straightforward. The most important fact to memorize for any recipe is: 1 US Cup = 16 US Tablespoons (tbsp) To find the number of tablespoons in any number of cups, you simply multiply the cup value by 16. Formula: Tablespoons = Cups × 16 Example: 0.25 cups × 16 = 4 tbsp Teaspoons to Tablespoons The second most important conversion to know is the relationship between tablespoons and teaspoons: 1 US Tablespoon (tbsp) = 3 US Teaspoons (tsp) This also means that 1 US Cup contains 48 teaspoons (16 tbsp × 3 tsp/tbsp). Cups to Tablespoons Quick Conversion Chart For your convenience, here is a chart for the most common baking and cooking measurements. Cups Tablespoons (tbsp) Teaspoons (tsp) 1 Cup 16 tbsp 48 tsp 3/4 Cup 12 tbsp 36 tsp 2/3 Cup 10 2/3 tbsp 32 tsp 1/2 Cup 8 tbsp 24 tsp 1/3 Cup 5 1/3 tbsp 16 tsp 1/4 Cup 4 tbsp 12 tsp 1/8 Cup 2 tbsp 6 tsp 1/16 Cup 1 tbsp 3 tsp Frequently Asked Questions (FAQ) How many tablespoons are in a cup? There are exactly **16 tablespoons** in 1 US Customary Cup. How many teaspoons are in a tablespoon? There are **3 teaspoons** in 1 tablespoon. How many tablespoons in 1/4 cup? There are **4 tablespoons** in 1/4 cup. (16

Home » Math & Conversions » Unit Conversions » Degrees to Radians Converter (with Pi) Degrees to Radians Converter A bidirectional converter for angles. Enter a value in any field to instantly convert between degrees and radians (as a decimal or in terms of π). Degrees (°) ≈ Radians (decimal) = Radians (in terms of π) π Calculation Steps: Enter a value to see the step-by-step conversion. How to Convert Degrees to Radians The conversion from degrees to radians is fundamental in trigonometry, calculus, and physics. The key is the relationship that **180 degrees is equal to π (pi) radians**. The Formula (Degrees to Radians) To convert from degrees to radians, you multiply the angle in degrees by the conversion factor `π

Home » Math & Conversions » Unit Conversions » Fahrenheit to Celsius (°F to °C) Converter Fahrenheit to Celsius (°F to °C) Converter A complete temperature conversion tool. Enter a value in any field to instantly convert between **Fahrenheit (°F)**, **Celsius (°C)**, and **Kelvin (K)**. Fahrenheit (°F) Celsius (°C) Kelvin (K) Calculation Steps: Enter a value to see the step-by-step conversion. How to Convert Fahrenheit to Celsius Converting from Fahrenheit (°F) to Celsius (°C) is a common requirement for anyone traveling or working with scientific data. The conversion can be done with a simple formula or a quick mental math trick. The Exact Formula To convert Fahrenheit to Celsius, you first subtract 32 from the temperature, then multiply the result by 5/9 (or ~0.555...). °C = (°F - 32) × 5/9 Example: Convert 70°F to °C. 1. 70 - 32 = 38 2. 38 × (5/9) = 21.11... °C The Easy Mental Math Trick For a quick estimate (especially for weather), you can use this simple trick: °C ≈ (°F - 30)

Home » Math & Conversions » Unit Conversions » Fahrenheit to Kelvin (°F to K) Converter Fahrenheit to Kelvin (°F to K) Converter A complete temperature conversion tool. Enter a value in any field to instantly convert between **Fahrenheit (°F)**, **Celsius (°C)**, and **Kelvin (K)**. Fahrenheit (°F) Celsius (°C) Kelvin (K) Calculation Steps: Enter a value to see the step-by-step conversion. Temperature Conversion Formulas Converting between Fahrenheit, Celsius, and Kelvin is essential in science and engineering. The most important thing to remember is that there is no direct formula for F to K; you must **convert to Celsius as an intermediate step**. Formula: Fahrenheit to Kelvin (°F → K) This is a two-step process: Step 1: Convert Fahrenheit to Celsius °C = (°F - 32) × 5/9 Step 2: Convert Celsius to Kelvin K = °C + 273.15 Combined Formula: K = ((°F - 32) × 5/9) + 273.15 Other Conversion Formulas Kelvin to Fahrenheit (K → °F): °F = (K - 273.15) × 9/5 + 32 Celsius to Fahrenheit (°C → °F): °F = (°C × 9/5) + 32 Fahrenheit to Celsius (°F → °C): °C = (°F - 32) × 5/9 Celsius to Kelvin (°C → K): K = °C + 273.15 Kelvin to Celsius (K → °C): °C = K - 273.15 Key Temperature Points Understanding these key reference points helps put the scales in context. Point Fahrenheit (°F) Celsius (°C) Kelvin (K) Absolute Zero -459.67 °F -273.15 °C 0 K Water Freezes 32 °F 0 °C 273.15 K Human Body (Avg) 98.6 °F 37 °C 310.15 K Water Boils 212 °F 100 °C 373.15 K Frequently Asked Questions (FAQ) What is the formula for Fahrenheit (°F) to Kelvin (K)? The formula is: K = (°F - 32) × 5/9 + 273.15 . You must first convert Fahrenheit to Celsius (by subtracting 32 and multiplying by 5/9), and then add 273.15 to the result. What is 0 Fahrenheit in Kelvin? 0 °F is equal to 255.372 K . 1. °C = (0 - 32) * 5/9 = -17.78 °C 2. K = -17.78 + 273.15 = 255.37 K What is Absolute Zero (0 K) in Fahrenheit? 0 K (Absolute Zero) is equal to -459.67 °F . This is the lowest possible temperature. Related Calculators Millimeters to Inches Converter Typography Unit Converter (px, pt, em, rem, %) Kilograms to Pounds Converter Centimeters to Inches Converter BTU to kWh Converter Angstroms to Nanometers Converter Kelvin to Fahrenheit Converter Degrees to Radians Converter Cups to Tablespoons Converter Barrels to Gallons Converter

Home » Math & Conversions » Unit Conversions » Feet to Meters (ft to m) Converter Feet to Meters (ft to m) Converter A complete length and height converter. Enter a value in any field to instantly convert between **feet/inches**, **meters**, and **centimeters**. Feet (ft) + Inches (in) ft in = Meters (m) = Centimeters (cm) How to Convert Feet and Inches to Meters Converting a measurement from feet (ft) and inches (in) to meters (m) is a simple two-step process. You cannot convert directly; you must first get a total in a single unit (like inches) and then convert that to meters. Step 1: Convert to Total Inches Multiply the number of feet by 12 (since there are 12 inches in a foot) and add the remaining inches. Formula: Total Inches = (Feet × 12) + Inches Step 2: Convert Total Inches to Meters The international standard is that 1 inch = 0.0254 meters. Multiply your total inches by 0.0254 to get the final answer in meters. Formula: Meters = Total Inches × 0.0254 Example: Convert 5' 10" to Meters Step 1: (5 ft × 12) + 10 in = 60 in + 10 in = 70 inches Step 2: 70 in × 0.0254 = 1.778 meters Conversion Formulas 1 Foot (ft) = 0.3048 Meters (m) 1 Inch (in) = 0.0254 Meters (m) 1 Meter (m) = 3.28084 Feet (ft) Height Conversion Chart (Feet/Inches to Meters) A quick reference chart for common human heights. Feet & Inches Meters (m) Centimeters (cm) 5' 0" 1.524 m 152.4 cm 5' 2" 1.575 m 157.5 cm 5' 4" 1.626 m 162.6 cm 5' 6" 1.676 m 167.6 cm 5' 8" 1.727 m 172.7 cm 5' 10" 1.778 m 177.8 cm 6' 0" 1.829 m 182.9 cm 6' 2" 1.880 m 188.0 cm 6' 4" 1.930 m 193.0 cm Frequently Asked Questions (FAQ) What is the formula for feet to meters? The exact formula is: Meters = Feet × 0.3048 . For example, 10 feet would be 10 * 0.3048 = 3.048 meters. How many feet are in a meter? There are approximately **3.28084 feet** in 1 meter. To convert meters to feet, you can divide the meter value by 0.3048. What is 5 ft 10 in meters (5'10")? 5 feet 10 inches is equal to **1.778 meters** (or 178 cm). The calculation is (5 feet * 12 inches/foot + 10 inches) = 70 inches. Then, 70 inches * 0.0254 m/inch = 1.778 m. Related Calculators Millimeters to Inches Converter Typography Unit Converter (px, pt, em, rem, %) Kilograms to Pounds Converter Centimeters to Inches Converter BTU to kWh Converter Angstroms to Nanometers Converter Kelvin to Fahrenheit Converter Degrees to Radians Converter Cups to Tablespoons Converter Barrels to Gallons Converter

Home » Math & Conversions » Unit Conversions » Joules to BTU (J to BTU) Converter Joules to BTU (J to BTU) Converter A dual-purpose scientific converter for **Energy (J, kJ, BTU)** and **Power (W, kW, BTU/hr)**. Enter a value in any field to convert instantly. Energy Conversion (J ↔ kJ ↔ BTU) Use this to convert total energy (e.g., thermal energy, battery capacity). Joules (J) Kilojoules (kJ) BTU (British Thermal Units) Power Conversion (W ↔ kW ↔ BTU/hr) Use this for appliance ratings (e.g., heaters, AC units). Note: **1 Watt = 1 Joule per second (J/s)**. Watts (W) Kilowatts (kW) BTU per Hour (BTU/hr) The Key Difference: Energy vs. Power The most common error in this conversion is confusing **Energy** (measured in Joules or BTU) with **Power** (measured in Watts or BTU/hr). Energy (Joule, BTU): This is a *total amount* of energy. Think of it like the amount of heat stored in a hot object. Power (Watt, BTU/hr): This is the *rate* at which energy is used or transferred. A 100-Watt light bulb uses 100 Joules of energy *every second*. This page provides both calculators. If you are converting a static energy value, use the first calculator. If you are converting an appliance rating (like a heater or AC unit), use the second calculator. Conversion Formulas Energy: 1 BTU = 1055.056 Joules (J) Power: 1 BTU/hr = 0.293071 Watts (W) To convert from Joules to BTU, you **divide by 1055.056** (or multiply by 0.0009478). To convert from Watts (J/s) to BTU/hr, you **multiply by 3.41214**. Common Conversion Chart Energy (Joule) Energy (BTU) Power (Watt) Power (BTU/hr) 1 J 0.000948 BTU 1 W 3.412 BTU/hr 1,000 J (1 kJ) 0.9478 BTU 1,000 W (1 kW) 3,412 BTU/hr 1,055 J 1 BTU 293 W 1,000 BTU/hr 10,000 J (10 kJ) 9.478 BTU 10,000 W (10 kW) 34,121 BTU/hr 1,000,000 J (1 MJ) 947.8 BTU 1,000,000 W (1 MW) 3,412,141 BTU/hr Frequently Asked Questions (FAQ) What is the formula for Joules (J) to BTU? The formula is: BTU = Joules × 0.000947817 . A simpler way is to divide the number of Joules by 1055.056. How many Joules are in 1 BTU? There are **1,055.056 Joules** (or 1.055 kilojoules) in 1 British Thermal Unit (BTU). Is a Watt the same as a Joule? No. A Joule (J) is a unit of energy. A Watt (W) is a unit of power (energy over time). They are related: **1 Watt = 1 Joule per second (J/s)**. Related Calculators BTU to kWh Converter kWh to BTU Converter Watts to Amps Calculator Kelvin to Fahrenheit Converter Angstroms to Nanometers Converter Millimeters to Inches Converter Kilograms to Pounds Converter Centimeters to Inches Converter Degrees to Radians Converter Cups to Tablespoons Converter

Home » Math & Conversions » Unit Conversions » Kelvin to Celsius (K to °C) Converter Kelvin to Celsius (K to °C) Converter A complete temperature conversion tool. Enter a value in any field to instantly convert between **Kelvin (K)**, **Celsius (°C)**, and **Fahrenheit (°F)**. Kelvin (K) Celsius (°C) Fahrenheit (°F) Calculation Steps: Enter a value to see the step-by-step conversion. How to Convert Kelvin to Celsius The conversion from Kelvin (K) to Celsius (°C) is the simplest temperature conversion because the *size* of one kelvin is identical to the size of one degree Celsius. The only difference is their starting point. The Formula (Kelvin to Celsius) To convert from Kelvin to Celsius, you simply subtract 273.15. °C = K - 273.15 Example: Convert the boiling point of water (373.15 K) to Celsius. 373.15 K - 273.15 = 100 °C The Formula (Celsius to Kelvin) To convert from Celsius to Kelvin, you add 273.15. K = °C + 273.15 What is Absolute Zero? The Kelvin scale is an "absolute" scale, meaning its zero point (0 K) is **Absolute Zero**. This is the theoretical temperature at which all atomic and molecular motion ceases. There is no temperature colder than 0 K, which is why the Kelvin scale has no negative numbers. Celsius and Fahrenheit are "relative" scales, as their zero points (0°C and 0°F) were set for convenience (e.g., the freezing point of water). Key Temperature Points Point Kelvin (K) Celsius (°C) Fahrenheit (°F) Absolute Zero 0 K -273.15 °C -459.67 °F Water Freezes 273.15 K 0 °C 32 °F Human Body (Avg) 310.15 K 37 °C 98.6 °F Water Boils 373.15 K 100 °C 212 °F Frequently Asked Questions (FAQ) What is the formula for Kelvin (K) to Celsius (°C)? The formula is: Celsius = Kelvin - 273.15 . (C = K - 273.15). What is 0 Kelvin in Celsius? 0 Kelvin (K) is equal to -273.15 °C . This is known as Absolute Zero. What is 0 Celsius in Kelvin? 0 °C (the freezing point of water) is equal to 273.15 K . Related Calculators Kelvin to Fahrenheit Converter Fahrenheit to Kelvin Converter Celsius to Fahrenheit Converter Fahrenheit to Celsius Converter Celsius to Kelvin Converter BTU to kWh Converter Millimeters to Inches Converter Kilograms to Pounds Converter Centimeters to Inches Converter

Home » Math & Conversions » Unit Conversions » Kilograms to Pounds (kg to lbs) Converter Kilograms to Pounds (kg to lbs) Converter Instantly convert kilograms (kg) to pounds (lbs) and ounces (oz). Enter a value in any field to see the live conversion. Kilograms (kg) = Pounds (decimal) = Pounds (lb) + Ounces (oz) lb oz How to Convert Kilograms to Pounds The conversion between kilograms (kg) and pounds (lbs) is one of the most common weight conversions. The key to understanding it is knowing the official conversion factor and a simple mental math trick. The Exact Formula By international agreement, one pound is defined as exactly 0.45359237 kilograms. The inverse of this, the conversion factor for kg to lbs, is: 1 kg = 2.2046226218 pounds (lbs) To convert any value from kilograms to pounds, you simply multiply the kilogram value by 2.2046226218. Example: Convert 70 kg to lbs. 70 kg × 2.2046226218 = 154.3235... lbs The "Mental Math" Trick (kg * 2.2) For a fast, everyday estimate, most people use a simpler factor: **2.2**. This is accurate enough for almost any non-scientific use. The "trick" is to multiply by 2, and then add 10% of that result (which is the same as multiplying by 0.2). Example: Convert 80 kg to lbs using the trick. Multiply by 2: 80 × 2 = 160 Take 10% of 80: 80 × 0.10 = 8 Add them together: 160 + 8 = 168 lbs (Note: The exact answer is 176.36 lbs. A more accurate trick is to "double it and add 10% of the *doubled* amount," i.e., 160 + 16 = 176 lbs. Our calculator uses the exact 2.2046... factor for perfect accuracy.) How to Convert to Pounds and Ounces To get pounds and ounces from a decimal, you must remember that there are 16 ounces (oz) in 1 pound (lb). Take the decimal value, e.g., 154.3235 lbs . The whole number is your pounds: 154 lbs . Take the remaining decimal (0.3235) and multiply it by 16: 0.3235 × 16 = 5.176 oz . So, 154.3235 lbs is equal to **154 lbs and 5.18 oz**. Kg to Lbs Quick Conversion Chart Kilograms (kg) Pounds (lbs) Pounds & Ounces (lb, oz) 1 kg 2.20 lbs 2 lbs, 3.27 oz 5 kg 11.02 lbs 11 lbs, 0.37 oz 10 kg 22.05 lbs 22 lbs, 0.74 oz 25 kg 55.12 lbs 55 lbs, 1.85 oz 50 kg 110.23 lbs 110 lbs, 3.70 oz 70 kg 154.32 lbs 154 lbs, 5.18 oz 100 kg 220.46 lbs 220 lbs, 7.40 oz Frequently Asked Questions (FAQ) Is a kilogram (kg) heavier than a pound (lb)? Yes, a kilogram is significantly heavier than a pound. 1 kilogram is equal to approximately 2.2046 pounds, so it's more than double the weight. How many kg in a pound? There are approximately 0.453592 kilograms in one pound. The formula to convert pounds to kg is: `Kilograms = Pounds

Home » Math & Conversions » Unit Conversions » Kilometers to Miles (km to miles) Converter Kilometers to Miles (km to miles) Converter A bidirectional converter for distance (km ↔ mi) and speed (kph ↔ mph). Enter a value in any field to convert instantly. Distance Conversion (km ↔ mi) Kilometers (km) Miles (mi) Speed Conversion (kph ↔ mph) For converting speeds, such as road signs or cycling data. Kilometers per Hour (kph) Miles per Hour (mph) How to Convert Kilometers to Miles Whether you're converting distance (km) or speed (kph), the conversion factor is the same. There are two main ways to do it: the exact formula for precision, and a simple mental math trick for quick estimates. The Exact Formula The internationally accepted conversion factor is: 1 Kilometer = 0.6213711922 Miles To convert from kilometers to miles, you simply multiply the kilometer value by 0.621371. Example: Convert 100 km to miles. 100 km × 0.621371 = 62.1371 miles The Easy Mental Math Trick (The "8 and 5" Rule) For a fast, on-the-go estimate (like seeing a road sign in Europe), this trick is extremely accurate. It's popular with cyclists and drivers. Miles ≈ (Kilometers

Home » Math & Conversions » Unit Conversions » kWh to BTU Converter kWh to BTU Converter A bidirectional converter for **Energy (kWh ↔ BTU)** and **Power (kW ↔ BTU/hr)**. Use the correct tool for your needs. Energy Conversion (kWh ↔ BTU) Use this to convert total energy (e.g., battery capacity, monthly bill). Kilowatt-Hours (kWh) BTU Power Conversion (kW ↔ BTU/hr) Use this for appliance ratings (e.g., air conditioners, heaters, stoves). Kilowatts (kW) BTU per Hour (BTU/hr) The Key Difference: Energy (kWh) vs. Power (kW) The most common error in this conversion is confusing **Energy** with **Power**. They are not the same. ENERGY (kWh and BTU): This is a *total amount* of energy. Think of it like the total fuel in your car's tank. You are billed by your utility company for the total energy (in kWh) you consume. POWER (kW and BTU/hr): This is the *rate* at which energy is used. Think of it as how *fast* your car burns fuel. An air conditioner's "10,000 BTU" rating is actually 10,000 BTU/hr (Power). This page provides both calculators. If you are converting a battery's capacity, use the Energy converter. If you are converting an appliance's rating, use the Power converter. Conversion Formulas Luckily, the conversion factor is numerically the same for both: Energy: 1 kWh = 3412.14 BTU Power: 1 kW = 3412.14 BTU/hr To convert from kWh to BTU, you **multiply by 3412.14**. To convert from BTU to kWh, you **divide by 3412.14** (or multiply by 0.000293071). Common Conversion Chart Kilowatts (kW) BTU per Hour (BTU/hr) Kilowatt-Hours (kWh) BTU 1 3,412 1 3,412 5 17,061 5 17,061 10 34,121 10 34,121 15 51,182 15 51,182 25 85,304 25 85,304 50 170,607 50 170,607 Frequently Asked Questions (FAQ) How many BTUs are in 1 kWh? There are **3,412.14 BTUs** in 1 kilowatt-hour (kWh). This is a conversion of *energy*. How many BTUs are in 1 kW? There are **3,412.14 BTU/hr** in 1 kilowatt (kW). This is a conversion of *power* (the rate of energy use). How do I convert my electric bill (kWh) to the BTU rating of my AC? You cannot convert them directly, as one is energy and one is power. However, you can find out how much your AC *cost* you. 1. Find your AC's power in kW: (BTU/hr)

Home » Math & Conversions » Unit Conversions » Liters to Gallons (L to gal) Converter Liters to Gallons (L to gal) Converter A complete liquid volume converter for kitchen, automotive, and scientific use. Enter a value in any field to instantly convert between all units (US and UK). Metric Units Liters (L) Milliliters (ml) US Customary Units US Gallons (gal) US Quarts (qt) US Pints (pt) US Cups US Fluid Ounces (fl oz) UK Imperial Units UK Gallons (gal) US Gallon vs. UK Gallon: A Critical Difference The most common error when converting "gallons" is not knowing which gallon you are using. The US and UK (Imperial) gallons are **not the same size**. 1 US Liquid Gallon = 3.7854 Liters 1 UK Imperial Gallon = 4.54609 Liters As you can see, the **UK gallon is approximately 20% larger** than the US gallon. This is critical for recipes, fuel economy calculations, and scientific measurements. Our calculator includes both to prevent this error. Conversion Formulas US Gallon to Liters: Liters = US Gallons × 3.7854 Liters to US Gallons: US Gallons = Liters × 0.264172 UK Gallon to Liters: Liters = UK Gallons × 4.54609 Liters to UK Gallons: UK Gallons = Liters × 0.219969 US Cooking Conversions For recipes, the smaller US units are all related: 1 US Gallon = 4 Quarts 1 US Quart = 2 Pints 1 US Pint = 2 Cups 1 US Cup = 8 Fluid Ounces 1 US Fluid Ounce = 2 Tablespoons Frequently Asked Questions (FAQ) How many liters are in a gallon? There are **3.7854 liters** in 1 **US Gallon**. There are **4.546 liters** in 1 **UK (Imperial) Gallon**. Is a US gallon the same as a UK gallon? No. The UK (Imperial) gallon is about 20% larger than the US gallon. (1 UK Gallon = 1.201 US Gallons). How many liters are in 5 gallons of gas? In the US, this would be 5 US Gallons, so: 5 × 3.7854 = 18.927 Liters . In the UK or Canada, this would be 5 Imperial Gallons, so: 5 × 4.546 = 22.73 Liters . Related Calculators Barrels to Gallons Converter Cups to Tablespoons Converter Tablespoons to Teaspoons Converter Milliliters to Cups Converter Kilograms to Pounds Converter Centimeters to Inches Converter Millimeters to Inches Converter BTU to kWh Converter kWh to BTU Converter Kelvin to Fahrenheit Converter

Home » Math & Conversions » Unit Conversions » Meters to Feet (m to ft) Converter Meters to Feet (m to ft) Converter A complete length and height converter. Enter a value in any field to instantly convert between **meters**, **centimeters**, and **feet/inches**. Meters (m) Centimeters (cm) = Feet (ft) + Inches (in) ft in How to Convert Meters to Feet and Inches Converting a measurement from meters (m) to feet (ft) and inches (in) is a common calculation for height, room dimensions, and more. It's a two-step process that requires first converting to total inches. Step 1: Convert Meters to Total Inches The international standard is that 1 inch is exactly 0.0254 meters. To convert, you divide your meter value by 0.0254 (or multiply by ~39.37). Formula: Total Inches = Meters

Home » Math & Conversions » Unit Conversions » Meters to Nanometers (m to nm) Converter Meters to Nanometers (m to nm) Converter A scientific calculator to convert meters (m), micrometers (µm), and nanometers (nm) instantly. Enter a value in any field to see all units update. Meters (m) Micrometers (µm or microns) Nanometers (nm) Scientific Notation (in Meters) - How to Convert Meters to Nanometers The conversion between meters (m) and nanometers (nm) is a simple factor of one billion (a 1 followed by nine zeros). This conversion is fundamental in fields like physics, nanotechnology, and optics. The Conversion Formula The relationship is based on the prefixes "micro-" (millionth) and "nano-" (billionth): 1 Meter (m) = 1,000,000,000 Nanometers (nm) 1 Meter (m) = 10 9 nm Therefore, to convert from meters to nanometers, you **multiply the meter value by 1,000,000,000**. To convert from nanometers to meters, you **divide the nanometer value by 1,000,000,000** (or multiply by 10 -9 ). The Scale of Metric Prefixes Understanding the "steps" from human scale (meters) down to the nano-scale is key. Each of these units is 1,000 times smaller than the one before it. Unit Symbol In Meters (Scientific) Example Meter m 1 m (10 0 m) Human height Millimeter mm 0.001 m (10 -3 m) Tip of a pencil Micrometer µm 0.000001 m (10 -6 m) Bacteria, water droplet Nanometer nm 0.000000001 m (10 -9 m) DNA strand, microchip circuits Angstrom Å 0.0000000001 m (10 -10 m) Atom size, chemical bonds Frequently Asked Questions (FAQ) How many nanometers are in a meter? There are **one billion (1,000,000,000)** nanometers in one meter. In scientific notation, this is written as 10 9 nm. What is 1 nanometer in meters? 1 nanometer (nm) is equal to one-billionth of a meter. As a decimal, this is 0.000000001 m. In scientific notation, it's written as 1×10 -9 m or 1e-9 m. What are nanometers used to measure? Nanometers are used to measure things on an atomic or molecular scale. This includes the wavelength of visible light (approx. 400-700 nm), the width of a strand of DNA (about 2.5 nm), and the tiny transistors on a computer chip (which can be smaller than 10 nm). Related Calculators Millimeters to Inches Converter Typography Unit Converter (px, pt, em, rem, %) Kilograms to Pounds Converter Centimeters to Inches Converter BTU to kWh Converter Angstroms to Nanometers Converter Kelvin to Fahrenheit Converter Degrees to Radians Converter Cups to Tablespoons Converter Barrels to Gallons Converter

Home » Math & Conversions » Unit Conversions » Meters to Yards (m to yd) Converter Meters to Yards (m to yd) Converter A complete length and measurement converter. Enter a value in any field to instantly convert between **meters/centimeters** and **yards/feet/inches**. Metric Units (m, cm) Meters Centimeters = Imperial Units (yd, ft, in) Yards Feet Inches How to Convert Meters to Yards The conversion between meters (m) and yards (yd) is based on a single, exact conversion factor defined by international agreement in 1959. The Exact Formula The relationship is fixed and exact: 1 Yard = 0.9144 Meters (Exactly) To convert from meters to yards, you **divide the meter value by 0.9144** (or multiply by ~1.09361). Formula: yd = m

Home » Math & Conversions » Unit Conversions » Miles to Kilometers (mi to km) Converter Miles to Kilometers (mi to km) Converter A bidirectional converter for distance (mi ↔ km) and speed (mph ↔ kph). Enter a value in any field to convert instantly. Distance Conversion (mi ↔ km) Miles (mi) Kilometers (km) Speed Conversion (mph ↔ kph) For converting speeds, such as road signs or cycling data. Miles per Hour (mph) Kilometers per Hour (kph) How to Convert Miles to Kilometers Whether you're converting distance (miles) or speed (mph), the conversion factor is the same. There are two main ways to do it: the exact formula for precision, and a simple mental math trick for quick estimates. The Exact Formula The internationally accepted conversion factor is: 1 Mile = 1.609344 Kilometers To convert from miles to kilometers, you simply multiply the mile value by 1.609344. Example: Convert 100 miles to km. 100 mi × 1.609344 = 160.9344 km The Easy Mental Math Trick (The "5 and 8" Rule) For a fast, on-the-go estimate (like seeing a road sign in the US), this trick is extremely accurate. It's popular with European drivers and cyclists. Kilometers ≈ (Miles

Home » Math & Conversions » Unit Conversions » Millimeters to Inches Converter Millimeters to Inches (mm to in) Converter Instantly convert between millimeters (mm) and inches (in). Enter a value in either field to see the live conversion as both a decimal and a fraction. Millimeters (mm) Inches (decimal) Inches (fraction) - How to Convert Millimeters to Inches Converting from millimeters (mm) to inches (in) is a common task in engineering, manufacturing, and everyday DIY projects. The process is straightforward and relies on a single conversion factor. The Conversion Formula The internationally accepted conversion factor is: 1 inch = 25.4 millimeters Therefore, to convert from millimeters to inches, you simply divide the number of millimeters by 25.4. Alternatively, you can multiply by the reciprocal, 0.0393701. Formula 1 (Division): Inches = Millimeters

Home » Math & Conversions » Unit Conversions » Nanometers to Angstroms (nm to Å) Converter Nanometers to Angstroms (nm to Å) Converter A scientific calculator to convert nanometers (nm), angstroms (Å), micrometers (µm), and meters (m) instantly. Enter a value in any field to see all units update. Nanometers (nm) Angstroms (Å) Micrometers (µm or microns) Meters (m) Scientific Notation (in Meters) - How to Convert Nanometers to Angstroms The conversion between nanometers (nm) and angstroms (Å) is a simple factor of 10. This conversion is extremely common in physics, chemistry, and materials science for measuring atomic-scale distances and wavelengths of light. The Conversion Formula The relationship is fixed and exact: 1 Nanometer (nm) = 10 Angstroms (Å) Therefore, to convert from nanometers to angstroms, you **multiply the nanometer value by 10**. Formula: Å = nm × 10 Example: 550 nm × 10 = 5500 Å To convert from angstroms to nanometers, you **divide the angstrom value by 10**. Formula: nm = Å

Home » Math & Conversions » Unit Conversions » Nanometers to Meters (nm to m) Converter Nanometers to Meters (nm to m) Converter A scientific calculator for the nano-scale. Instantly convert between **nanometers (nm)**, **micrometers (µm)**, **millimeters (mm)**, and **meters (m)** by entering a value in any field. Nanometers (nm) Micrometers (µm or microns) Millimeters (mm) Meters (m) Nanometers in Scientific Notation - How to Convert Nanometers to Meters The conversion between nanometers (nm) and meters (m) is a simple base-10 relationship. A nanometer is one-billionth of a meter. The Conversion Formula The relationship is defined as: 1 Meter (m) = 1,000,000,000 Nanometers (nm) 1 Nanometer (nm) = 0.000000001 Meters (m) In scientific notation, this is much easier to write: 1 nm = 1×10 -9 m Therefore, to convert from nanometers to meters, you **divide the nanometer value by 1,000,000,000** (one billion). Formula: m = nm

Home » Math & Conversions » Unit Conversions » Nanometers to Micrometers (nm to µm) Converter Nanometers to Micrometers (nm to µm) Converter A scientific calculator for the nano-scale. Instantly convert between **nanometers (nm)**, **micrometers (µm)**, **millimeters (mm)**, and **meters (m)** by entering a value in any field. Nanometers (nm) Micrometers (µm or microns) Millimeters (mm) Meters (m) Nanometers in Scientific Notation (nm) - How to Convert Nanometers to Micrometers The conversion between nanometers (nm) and micrometers (µm) is a simple factor of 1000. These units are essential in science and engineering for measuring microscopic and nanoscopic objects. The Conversion Formula The relationship is based on the metric prefixes "micro-" (one-millionth) and "nano-" (one-billionth). 1 Micrometer (µm) = 1,000 Nanometers (nm) Therefore, to convert from nanometers to micrometers, you **divide the nanometer value by 1,000**. Formula: µm = nm

Home » Math & Conversions » Unit Conversions » Ounces to Grams (oz to g) Converter Ounces to Grams (oz to g) Converter A fast, bidirectional converter for **mass (weight)**. Enter a value in either field to convert between ounces (oz) and grams (g). Weight/Mass Conversion (oz ↔ g) Ounces (oz) Grams (g) Bonus: Volume Converter (fl oz ↔ ml) Confused? You might be looking for **fluid ounces (volume)**, not ounces (weight). Use this tool to convert liquid measurements. US Fluid Ounces (fl oz) Milliliters (ml) Ounces vs. Fluid Ounces: A Critical Difference The word "ounce" is one of the most confusing units. It's crucial to know if you are measuring **weight (mass)** or **volume**. They are not the same! Ounce (oz): This is a unit of **WEIGHT/MASS**. It is used for dry ingredients like flour, sugar, or nuts. The formula is 1 oz = 28.34952 grams . Fluid Ounce (fl oz): This is a unit of **VOLUME**. It is used for liquids like water, milk, or oil. The formula is 1 US fl oz = 29.57353 milliliters . A "cup of flour" and a "cup of water" have the same **volume** (8 fl oz) but very different **weights** (mass). Our first calculator is for weight (oz to g), and the second is for volume (fl oz to ml). Conversion Formulas Mass: 1 Ounce (oz) = 28.34952 Grams (g) Volume: 1 US Fluid Ounce (fl oz) = 29.57353 Milliliters (ml) Ounces to Grams Quick Conversion Chart (Mass) Ounces (oz) Grams (g) - approx. 0.25 oz (1/4 oz) 7.09 g 0.5 oz (1/2 oz) 14.17 g 1 oz 28.35 g 4 oz 113.4 g 8 oz (1/2 lb) 226.8 g 16 oz (1 lb) 453.59 g Frequently Asked Questions (FAQ) How many grams are in 1 ounce? There are **28.34952 grams** in 1 standard (avoirdupois) ounce. For cooking, this is often rounded to 28.35 g. How many grams are in 8 oz? You multiply 8 by 28.34952, which equals **226.796 grams**. Is a fluid ounce (fl oz) the same as an ounce (oz)? No. An ounce (oz) measures weight/mass, while a fluid ounce (fl oz) measures volume. 1 cup of water has a volume of 8 fl oz and a weight of about 8.3 oz. 1 cup of flour has the same 8 fl oz volume but only weighs about 4.25 oz. Related Calculators Millimeters to Inches Converter Typography Unit Converter (px, pt, em, rem, %) Kilograms to Pounds Converter Centimeters to Inches Converter BTU to kWh Converter Angstroms to Nanometers Converter Kelvin to Fahrenheit Converter Degrees to Radians Converter Cups to Tablespoons Converter Barrels to Gallons Converter

Home » Math & Conversions » Unit Conversions » Pascals to PSI Converter Pascals (Pa) to PSI Converter A multi-unit pressure converter. Enter a value in any field to instantly convert between Pascals (Pa), Kilopascals (kPa), Megapascals (MPa), and Pounds per Square Inch (PSI). Pounds per Square Inch (PSI) Kilopascals (kPa) Megapascals (MPa) Pascals (Pa) Scientific Notation (in Pascals) - How to Convert Pascals to PSI The conversion between Pascals (Pa), the SI unit of pressure, and Pounds per Square Inch (PSI), the Imperial unit, is a common task in engineering and science. The most important conversion factor to remember is: 1 PSI = 6,894.757 Pascals (Pa) Formulas Used Pascals to PSI: PSI = Pa

Home » Math & Conversions » Unit Conversions » Picometers to Angstroms (pm to Å) Converter Picometers to Angstroms (pm to Å) Converter A scientific calculator for the atomic scale. Instantly convert between **picometers (pm)**, **angstroms (Å)**, **nanometers (nm)**, and **meters (m)** by entering a value in any field. Picometers (pm) Angstroms (Å) Nanometers (nm) Meters (m) Scientific Notation (in Meters) - How to Convert Picometers to Angstroms The conversion between picometers (pm) and angstroms (Å) is a simple, direct factor of 100. Both are used extensively in chemistry and physics to describe distances at the atomic and molecular level. The Conversion Formula The relationship is defined as: 1 Angstrom (Å) = 100 Picometers (pm) Therefore, to convert from picometers to angstroms, you **divide the picometer value by 100**. Formula: Å = pm

Home » Math & Conversions » Unit Conversions » Pounds to Kilograms (lbs to kg) Converter Pounds to Kilograms (lbs to kg) Converter A fast, bidirectional converter for pounds, ounces, and kilograms. Enter a value in any field to see all values update instantly. Pounds (lb) + Ounces (oz) lb oz = Pounds (decimal) = Kilograms (kg) How to Convert Pounds to Kilograms Converting pounds (lbs) to kilograms (kg) is a common conversion, especially for body weight, international travel, and science. The process is simple, whether you use the exact formula or a quick mental math trick. The Exact Formula By international agreement (the International Yard and Pound Agreement of 1959), one pound is defined as exactly **0.45359237 kilograms**. 1 lb = 0.45359237 kg To convert any value from pounds to kilograms, you simply multiply the pound value by 0.45359237. Example: Convert 150 lbs to kg. 150 lbs × 0.45359237 = 68.03885... kg The Mental Math Trick (How to Convert in Your Head) For a fast, everyday estimate, the easiest way is to divide by **2.2**. A simple trick for this (as mentioned on Reddit and other forums) is to **"divide by 2, then subtract 10%".** Example: Convert 180 lbs to kg using the trick. Divide by 2: 180

Home » Math & Conversions » Unit Conversions » PSI to Bar Converter PSI to Bar Converter A complete pressure unit conversion tool. Enter a value in any field to instantly convert between **PSI**, **Bar**, **Kilopascals (kPa)**, and **Atmospheres (atm)**. Pounds per Square Inch (PSI) Bar Kilopascals (kPa) Atmospheres (atm) How to Convert PSI to Bar Converting from Pounds per Square Inch (PSI) to bar is a common task for checking tire pressure or in engineering applications. The conversion is straightforward. The Conversion Formula The relationship between PSI and bar is derived from their definitions in Pascals: 1 bar = 100,000 Pascals (Pa) 1 PSI = 6,894.757 Pascals (Pa) From this, we get the direct conversion factors: PSI to Bar: bar = PSI × 0.06894757 (or divide by 14.50377) Bar to PSI: PSI = bar × 14.50377 PSI vs Bar for Tire Pressure This is the most common reason for this conversion. Different regions and manufacturers use different units: PSI is standard in the **United States**. A typical car tire pressure is 30-35 PSI. Bar is the standard metric unit in **Europe and the UK**. A typical car tire pressure is 2.1-2.5 bar. Common Tire Pressure Conversion Chart PSI Bar (approx.) kPa (approx.) 28 psi 1.93 bar 193 kPa 30 psi 2.07 bar 207 kPa 32 psi 2.21 bar 221 kPa 35 psi 2.41 bar 241 kPa 38 psi 2.62 bar 262 kPa 40 psi 2.76 bar 276 kPa Frequently Asked Questions (FAQ) What is the formula for PSI to bar? The formula is: bar = PSI

Home » Math & Conversions » Unit Conversions » PSI to Pascals Converter PSI to Pascals (Pa) Converter A multi-unit pressure converter. Enter a value in any field to instantly convert between **PSI**, **Pascals (Pa)**, **Kilopascals (kPa)**, and **Megapascals (MPa)**. Pounds per Square Inch (PSI) Megapascals (MPa) Kilopascals (kPa) Pascals (Pa) PSI in Scientific Notation - How to Convert PSI to Pascals Converting Pounds per Square Inch (PSI) to Pascals (Pa) is a common calculation in engineering and science. The Pascal (Pa) is the SI base unit for pressure, but it is a very small unit, so **Kilopascals (kPa)** and **Megapascals (MPa)** are more frequently used. The Conversion Formula The exact international conversion factor is: 1 PSI = 6,894.75729 Pascals (Pa) To convert from PSI to Pascals, you **multiply the PSI value by 6,894.757**. Formula: Pa = PSI × 6894.757 Example: 32 PSI × 6894.757 = 220,632.23 Pa Converting to kPa and MPa Because the Pascal value is so large, it's more practical to convert to kilopascals (1,000 Pa) or megapascals (1,000,000 Pa). PSI to Kilopascals (kPa): kPa = PSI × 6.89476 PSI to Megapascals (MPa): MPa = PSI × 0.00689476 Using our 32 PSI example: 32 * 6.89476 = 220.63 kPa , or 0.22 MPa . Common Pressure Reference Points Measurement PSI (approx.) Pascals (Pa) Kilopascals (kPa) Standard Atmosphere 14.696 psi 101,325 Pa 101.325 kPa 1 bar 14.504 psi 100,000 Pa 100 kPa Car Tire Pressure 32 psi 220,632 Pa 220.6 kPa Residential Water Pressure 50 psi 344,738 Pa 344.7 kPa Frequently Asked Questions (FAQ) What is the formula for PSI to Pascals (Pa)? The formula is: Pascals = PSI × 6894.757 . How many Pascals are in 1 PSI? There are **6,894.757 Pascals** in 1 Pound per Square Inch (PSI). This is often rounded to 6,895. How many Pascals in 1 kPa? There are exactly 1,000 Pascals (Pa) in 1 Kilopascal (kPa). Related Calculators PSI to Bar Converter Atmospheres to PSI Converter Pascals to PSI Converter kWh to BTU Converter Kelvin to Fahrenheit Converter Fahrenheit to Kelvin Converter Angstroms to Nanometers Converter Nanometers to Meters Converter Millimeters to Inches Converter Kilograms to Pounds Converter

Home » Math & Conversions » Unit Conversions » Square Meters to Square Feet (m² to ft²) Converter Square Meters to Square Feet (m² to ft²) Converter A complete area converter for real estate and home projects. Use the 'Area from Dimensions' tool for floor plans, or the multi-unit converter for `m²`, `ft²`, `yd²`, and `acres`. Calculate Area from Dimensions Enter room or land dimensions in meters to instantly find the area in both m² and ft². Length (meters) Width (meters) Area (Square Meters) Area (Square Feet) Multi-Unit Area Converter Enter a value in any field to convert between all units. Square Meters (m²) Square Feet (ft²) Square Yards (yd²) Acres How to Convert Square Meters to Square Feet The conversion between square meters (m²) and square feet (ft²) is a common calculation for real estate, landscaping, and home projects. It's important to remember that you are converting **area**, not length. The Conversion Formula You can't just use the conversion for meters to feet. You must use the **area** conversion factor. Length Factor: 1 meter = 3.28084 feet Area Factor: 1 square meter = 1 m × 1 m ...which means: 1 m² = 3.28084 ft × 3.28084 ft = 10.76391 ft² Square Feet = Square Meters × 10.7639 Example: To convert a 100 m² apartment to square feet: 100 m² × 10.7639 = 1076.39 ft² How to Convert Square Feet to Square Meters To convert the other way, you divide by 10.7639, or multiply by the inverse factor (0.092903). Square Meters = Square Feet × 0.092903 Common Area Conversions (m² to ft²) Square Meters (m²) Square Feet (ft²) Approx. Common Use 1 m² 10.76 ft² Large closet 10 m² 107.64 ft² Small bedroom (e.g., 10x10 ft) 25 m² 269.10 ft² Large living room 50 m² 538.19 ft² Small apartment 100 m² 1076.39 ft² Average 2-bed apartment 150 m² 1614.59 ft² Small family home Frequently Asked Questions (FAQ) What is the formula for m² to ft²? The formula is: Square Feet = Square Meters × 10.7639 . How many square feet are in a square meter? There are **10.7639 square feet** in 1 square meter. Which is bigger, a square meter or a square foot? A **square meter is much larger** than a square foot. It is over 10 times bigger. Related Calculators Feet to Meters Converter Centimeters to Inches Converter Millimeters to Inches Converter Kilometers to Miles Converter Miles to Kilometers Converter Kilograms to Pounds Converter Pounds to Kilograms Converter BTU to kWh Converter Kelvin to Fahrenheit Converter Fahrenheit to Kelvin Converter

Home » Math & Conversions » Unit Conversions » Tablespoons to Teaspoons Converter Tablespoons to Teaspoons (tbsp to tsp) Converter Your complete kitchen conversion tool. Enter a value in any field to instantly convert between **Cups**, **Fluid Ounces (fl oz)**, **Tablespoons (tbsp)**, and **Teaspoons (tsp)**. Cups (US Customary) Fluid Ounces (fl oz) Tablespoons (tbsp) Teaspoons (tsp) Kitchen Conversion Basics In cooking and baking, especially in the US, measurements are based on a standard hierarchy. Memorizing just three simple rules will allow you to convert almost any recipe. 1 Tablespoon = 3 Teaspoons 1 Fluid Ounce = 2 Tablespoons 1 Cup = 16 Tablespoons From these three rules, all other conversions can be found: 1 Cup = 16 Tablespoons = **48 Teaspoons** 1 Cup = 8 Fluid Ounces = **16 Tablespoons** 1/2 Cup = 8 Tablespoons = **24 Teaspoons** 1/4 Cup = 4 Tablespoons = **12 Teaspoons** Cooking Conversion Chart (US) For convenience, here is a chart for the most common measurements you'll find in recipes. Measurement Cups Fluid Ounces Tablespoons Teaspoons 1 Cup 1 8 fl oz 16 tbsp 48 tsp 1/2 Cup 1/2 4 fl oz 8 tbsp 24 tsp 1/4 Cup 1/4 2 fl oz 4 tbsp 12 tsp 1 Tablespoon 1/16 0.5 fl oz 1 tbsp 3 tsp 1 Teaspoon 1/48 ~0.167 fl oz 1/3 tbsp 1 tsp Frequently Asked Questions (FAQ) How many teaspoons are in a tablespoon? There are **3 teaspoons (tsp)** in 1 tablespoon (tbsp). This is the most fundamental conversion in US kitchen measurements. How many tablespoons are in a cup? There are **16 tablespoons (tbsp)** in 1 US Customary Cup. Is a UK tablespoon the same as a US tablespoon? No, they are different, which can be confusing. • **US Tablespoon:** 14.79 mL (defined as 1/2 US fl oz) • **UK/Imperial & Metric Tablespoon:** 15 mL (this is the modern standard in the UK, Canada, and Australia). For most recipes, this small difference is not critical, but for precise baking, it can matter. Related Calculators Millimeters to Inches Converter Typography Unit Converter (px, pt, em, rem, %) Kilograms to Pounds Converter Centimeters to Inches Converter BTU to kWh Converter Angstroms to Nanometers Converter Kelvin to Fahrenheit Converter Degrees to Radians Converter Cups to Tablespoons Converter Barrels to Gallons Converter

Home » Math & Conversions » Unit Conversions » Typography Converter (Fancy Text & CSS Units) Typography Converter (Fancy Text & CSS Units) Use our all-in-one typography tool to generate **fancy text fonts** for your social media profiles (Instagram, Twitter, Discord) or to **convert technical CSS units** (px, pt, em, rem) for web design. Fancy Text Generator (for Social Media) Enter your text here: Click any box below to copy the style. CSS Typography Unit Converter Base Font Size (px) This is your root () font-size. 1rem will be equal to this value. Pixels (px) Points (pt) EMs (em) REMs (rem) Percent (%) About Typography Converters The term "typography converter" can mean two very different things, both of which this page provides. 1. Fancy Text vs. Real Fonts Tools like LingoJam or Picsart are "fancy text generators." They don't create *fonts* that you can install. Instead, they use **Unicode**, a system that contains thousands of special characters, including symbols that look like letters from different alphabets (like cursive, Fraktur, or bold sans-serif). When you copy 𝓯𝓪𝓷𝓬𝔂 𝓽𝓮𝔁𝓽, you are not copying a font. You are copying a sequence of specific Unicode symbols. This is why you can paste it into social media bios (like Instagram or Twitter), as these platforms support Unicode but not custom fonts. 2. CSS Units: Absolute vs. Relative For web designers, typography conversion is about CSS units. These units fall into two categories: Absolute Units: These units are fixed and do not change. Pixels (px) are the most common absolute unit for screens. Points (pt) are from print design (1pt = 1/72 of an inch) and should generally be avoided for web use. Relative Units: These units scale based on another value. This is crucial for creating responsive and accessible websites. REM (Root EM): The best and most popular unit. It is relative *only* to the root font size (the `` element), which is typically 16px by default. EM: Relative to the font size of its *direct parent element*. This can become very complicated and lead to unexpected results (e.g., 1.2em inside 1.2em). Percent (%): Behaves identically to 'em'. Frequently Asked Questions (FAQ) What is the best unit for web typography? Most modern web developers and accessibility experts recommend using **rem** units for `font-size`, `padding`, `margin`, and `width`. By basing all of your site's sizing on the root font size, you allow users to scale the *entire website* up or down perfectly just by changing their browser's default font size setting. How do you convert px to rem? The formula is simple: `rem = px

Home » Math & Conversions » Unit Conversions » Yards to Meters (yd to m) Converter Yards to Meters (yd to m) Converter A complete length and measurement converter. Enter a value in any field to instantly convert between **meters/centimeters** and **yards/feet/inches**. Imperial Units (yd, ft, in) Yards Feet Inches = Metric Units (m, cm) Meters Centimeters How to Convert Yards to Meters Converting yards (yd) to meters (m) is based on a single, exact conversion factor defined by international agreement in 1959. The Exact Formula The relationship is fixed and exact: 1 Yard = 0.9144 Meters (Exactly) To convert from yards to meters, you **multiply the yard value by 0.9144**. Formula: m = yd × 0.9144 Example: 100 yd × 0.9144 = 91.44 m To convert from meters to yards, you **divide the meter value by 0.9144** (or multiply by ~1.09361). Formula: yd = m

Home » Science Calculators » Biology Biology Calculators Streamline genetics, microbiology, ecology, and biochemistry workflows with calculators that document formulas, guidance, and accepted references. DNA → protein translator Convert DNA or RNA sequences into amino acid chains with codon table annotations. DNA concentration calculator Normalize A260 readings to ng/µL, molarity, and copy number with dilution options. Hemocytometer counter Convert grid counts into cells per mL with dilution factors and viability tracking. Cell doubling time Estimate population doubling for exponential growth or decay phases. Bacterial growth curve Model lag, exponential, stationary, and death phases using OD or cell counts. Allele frequency calculator Track allele and genotype frequencies with Hardy–Weinberg and selection options. Hardy–weinberg equilibrium Evaluate equilibrium assumptions, chi-square tests, and allele shifts. Punnett square generator Visualize monohybrid or dihybrid crosses with genotype and phenotype ratios. Restriction digest planner Design single or double digests with enzyme compatibility and fragment prediction. Michaelis–menten calculator Fit enzyme kinetics, Km, Vmax, and Lineweaver–Burk transformations. Lotka–volterra simulator Explore predator–prey population dynamics with adjustable interaction rates. Island biogeography Estimate equilibrium species richness from colonization and extinction rates. Niche breadth calculator Measure ecological niche width using Levin’s or Shannon diversity indices. Population pyramid generator Create age–sex demographic charts from raw counts or normalized proportions. Life table calculator Compute survivorship (lx), mortality (qx), and life expectancy (ex) metrics. Fick's law diffusion Calculate flux through membranes using concentration gradients and diffusion areas. Related categories Science Calculators Health & Fitness Math & Conversions Chemistry Tools

Home » Science Calculators » Chemistry Chemistry Calculators Plan reactions, prep solutions, and interpret lab results with calculators covering stoichiometry, equilibrium, spectroscopy, and electrochemistry. Molecular weight calculator Parse chemical formulas, hydrates, and nested groups to compute molar mass and composition. Molar mass helper Break down empirical or molecular formulas and export atomic contribution tables. Stoichiometry calculator Convert between moles, grams, and particles with balanced equation guidance. Limiting reactant Identify the limiting reagent, theoretical yield, and excess remaining for reactions. Percent yield Compare actual versus theoretical yield with limiting reactant integration. Grams to moles Convert sample mass into amount of substance using molar mass lookups. Moles to grams Scale products and reactants with automatic molar mass lookup and rounding. Equation balancer Balance chemical equations, format states, and download LaTeX-friendly output. Empirical formula Find simplest whole-number ratios from percent composition or elemental mass. Percent composition Calculate mass percent of each element in a compound or mixture. Molarity calculator Compute solution molarity, required solute mass, or solvent volume interchangeably. Solution dilution Apply C1V1 = C2V2 for quick dilution series or single-step adjustments. Serial dilution planner Design multi-step dilutions with tube-by-tube volumes and cumulative factors. Titration curve Simulate acid-base titrations with equivalence points, buffer regions, and charts. Ph calculator Determine pH, pOH, [H+], and [OH-] for strong acids/bases and dilute solutions. Henderson–hasselbalch Model buffer pH from pKa and conjugate acid-base ratios with unit conversions. Beer–lambert law Relate absorbance, concentration, path length, and molar absorptivity. Arrhenius equation Estimate reaction rates or activation energy from temperature-dependent kinetics. Ideal gas law Solve PV = nRT for any variable with unit-aware conversions and STP presets. Van 't hoff factor Evaluate colligative properties, dissociation, and freezing point depression impacts. Nernst equation Predict electrode potentials under non-standard conditions with log-based outputs. Electronegativity difference Assess bond polarity, ionic character, and dominant interaction type. Gibbs free energy Relate Delta G, Delta H, Delta S, and temperature to determine reaction spontaneity. Isotope abundance Derive average atomic mass from isotope masses and fractional abundances. Born–haber cycle Assemble lattice energies from enthalpy changes in ionic compound formation. Percent error Quantify analytical accuracy with signed or absolute percent error outputs. Fick's law of diffusion calculator | life and earth sciences Calculate diffusion rates using Fick Related categories Science Calculators Physics Calculators Math & Conversions Biology Tools

Home » Science Calculators » Physics Physics Calculators Solve mechanics, electromagnetism, relativity, and optics problems with calculators that include formulas, unit support, and contextual guidance. Acceleration calculator Determine constant acceleration from velocity change, distance, or time inputs. Force calculator Apply Newton’s second law with vector-aware force, mass, and acceleration outputs. Speed, distance & time Solve kinematics problems with interchangeable speed, distance, and time variables. Momentum calculator Compute linear momentum, impulse, and collision outcomes with unit conversions. Work calculator Evaluate mechanical work, force, displacement, and angle relationships. Power calculator Derive mechanical and electrical power from work rate, torque, or energy change. Kinetic energy calculator Calculate translational kinetic energy, solve for velocity, or compare masses. Potential energy calculator Model gravitational, elastic, and electric potential energy scenarios. Torque calculator Find torque from applied force, lever arm, and angle in static or dynamic cases. Centripetal force Analyze circular motion with mass, radius, angular velocity, and frequency inputs. Density calculator Convert between mass density, specific gravity, and volumetric units. Pressure calculator Relate force, area, and fluid columns with scientific and engineering units. Diffraction grating Predict maxima angles, order spacing, and wavelength from grating setups. Single-slit diffraction Model interference minima and maxima for laser and aperture experiments. Photoelectric effect Calculate photon energy, stopping potential, and kinetic energy thresholds. Planck's law calculator Evaluate blackbody spectra, radiance peaks, and wavelength-temperature relations. Relativistic time dilation Compute Lorentz factors, proper time, and velocity fractions near light speed. Relativistic velocity addition Combine velocities using special relativity instead of classical vector sums. Relativistic kinetic energy Compare classical and relativistic kinetic energy across particle energies. Relativistic mass calculator Compute relativistic mass increase with Lorentz factors and velocity inputs. Nuclear q-value calculator Assess energy released or absorbed in nuclear reactions from mass deficits. Uncertainty principle Relate momentum, position, energy, and time uncertainties in quantum systems. Hydrostatic pressure calculator Calculate hydrostatic pressure with ease using this precise, professional-grade calculator for fluid mechanics. Thin lens equation calculator Use this Thin Lens Equation Calculator to determine the relationship between object distance, image distance, and focal length in optics. Related categories Science Calculators Engineering Tools Math & Conversions Lifestyle & Everyday

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Math & Conversions

Math & Conversions Angle Calculator Angle Calculator (Right Triangle Solver) This calculator solves for the missing angles and sides of any right-angled triangle. Enter the minimum two known values (two sides, or one side and one acute angle) in the corresponding fields below. Enter Known Values (Minimum 2) Side a (Opposite $\alpha$) Side b (Opposite $\beta$) Side c (Hypotenuse) Angle $\alpha$ (degrees) Angle $\beta$ (degrees) Angle $\gamma = 90^\circ$ Solve Triangle Solved Triangle Side Results Angle Results Step-by-Step Solution The Core Formulas of Right Triangle Trigonometry A right triangle is fully defined by three key relationships: 1. Pythagorean Theorem (Sides) Used to find the length of any side when the other two sides are known: $$a^2 + b^2 = c^2$$ Where $c$ is the hypotenuse (the side opposite the $90^\circ$ angle), and $a$ and $b$ are the legs. 2. Angle Sum Rule The sum of the internal angles of any triangle is $180^\circ$ (or $\pi$ radians). Since the right angle $\gamma$ is $90^\circ$, the two acute angles ($\alpha$ and $\beta$) must sum to $90^\circ$: $$\alpha + \beta + 90^\circ = 180^\circ \quad \text{or} \quad \alpha + \beta = 90^\circ$$ 3. SOHCAHTOA (Sides and Angles) The trigonometric ratios relate the angles to the ratio of side lengths: $$\sin(\theta) = \frac{\text{Opposite}}{\text{Hypotenuse}} \quad \cos(\theta) = \frac{\text{Adjacent}}{\text{Hypotenuse}} \quad \tan(\theta) = \frac{\text{Opposite}}{\text{Adjacent}}$$ To find a missing angle ($\theta$) when all sides are known, we use the inverse functions (arcsin, arccos, arctan): $\theta = \sin^{-1}(\frac{O}{H})$. Frequently Asked Questions (FAQ) What is SOHCAHTOA? SOHCAHTOA is a mnemonic device used to remember the three basic trigonometric ratios in a right triangle: SOH: $\sin(\theta) = \frac{\text{Opposite}}{\text{Hypotenuse}}$ CAH: $\cos(\theta) = \frac{\text{Adjacent}}{\text{Hypotenuse}}$ TOA: $\tan(\theta) = \frac{\text{Opposite}}{\text{Adjacent}}$ What is the sum of angles in a triangle? The sum of the interior angles of any planar triangle, including a right triangle, is always $180$ degrees ($\pi$ radians). What is the Law of Sines and Law of Cosines? The Law of Sines and Law of Cosines are trigonometric rules used to solve **non-right** triangles (oblique triangles). For right triangles, SOHCAHTOA and the Pythagorean theorem are generally sufficient and simpler. What is the required minimum input to solve a right triangle? To solve a right triangle (i.e., find all missing sides and angles), you need to know exactly two non-angle parts (two sides) or one side and one acute angle. Knowing the right angle ($90^\circ$) is already assumed. Trigonometry Basics Pythagorean Theorem $$c^2 = a^2 + b^2$$ Tangent (TOA) $$\tan(\alpha) = \frac{a}{b}$$ Inverse Sine (SOH) $$\alpha = \sin^{-1}\left(\frac{a}{c}\right)$$ Related Geometry Tools Right Triangle Calculator Pythagorean Theorem Calculator Triangle Calculator (General) Law of Sines Calculator Law of Cosines Calculator Radians to Degrees Converter

Math & Conversions Arc Length Calculator Arc Length Calculator ($L$) Calculate the length of the arc ($L$) along the circumference of a circle, determined by its radius ($r$) and the central angle ($\theta$). Enter your values below and select whether the angle is in degrees or radians. Enter Dimensions Radius ($r$) Central Angle ($\theta$) Degrees ($^\circ$) Radians Calculate Arc Length Key Results Arc Length ($L$) Circumference ($C$) Sector Area ($\mathcal{A}$) Step-by-Step Solution Formulas for Arc Length The arc length is defined as a fraction of the circle's total circumference. The formula used depends on whether the central angle ($\theta$) is measured in radians or degrees. 1. Arc Length using Radians ($\theta$) The simplest and most fundamental relationship: the arc length ($L$) is the product of the radius and the angle (in radians). $$L = r \theta$$ This is the preferred formula for scientific and engineering calculations because of its simplicity. 2. Arc Length using Degrees ($\theta^\circ$) The formula for degrees explicitly shows the ratio of the partial angle to the full circle: $$L = 2\pi r \left( \frac{\theta}{360^\circ} \right)$$ Radians Conversion If you start with degrees, you must first convert the angle to radians to use the simpler formula: $$\theta_{\text{radians}} = \theta_{\text{degrees}} \times \frac{\pi}{180^\circ}$$ Arc Length vs. Chord Length It is important to distinguish the arc length from the chord length (which is the straight-line distance between the two endpoints of the arc). The arc length is always greater than the chord length, except when the angle is $0^\circ$ or $360^\circ$, where they are equal. Frequently Asked Questions (FAQ) What is the formula for arc length in radians? When the central angle ($\theta$) is measured in radians, the arc length ($L$) is calculated by the formula $L = r \\theta$. What is the formula for arc length in degrees? When the central angle ($\theta$) is measured in degrees, the arc length ($L$) is calculated as a fraction of the total circumference: $L = 2\pi r \left( \\frac{\\theta}{360^\\circ} \\right)$. What is the difference between arc length and chord length? The **arc length** is the measurement of the curved path along the circumference between the two endpoints of the radii. The **chord length** is the straight-line distance connecting those same two endpoints. How does this differ from the arc length formula in Calculus? In Calculus, the arc length formula is used to find the length of a curve defined by a function $f(x)$ over an interval $[a, b]$, using an integral: $L = \int_{a}^{b} \sqrt{1 + (f'(x))^2} dx$. This geometric calculator only addresses the arc length of a perfect circle. Arc Length Key Formulas Length (Radians) $$L = r \theta$$ Length (Degrees) $$L = C \frac{\theta}{360^\circ}$$ Sector Area $$\mathcal{A} = \frac{1}{2} r L$$ Related Circle Tools Sector Area Calculator Circle Calculator (Area, Circumference) Central Angle Calculator Degrees to Radians Converter Chord Length Calculator

Math & Conversions Circle Calculator Circle Calculator: Area, Circumference, Diameter, Radius This calculator can solve for the four main properties of a circle. Enter a value for **any one** of the properties below (Radius, Diameter, Circumference, or Area), and the tool will instantly derive all the others. Geometric Properties Radius ($r$) Diameter ($d$) Circumference ($C$) Area ($A$) Calculate Circle Results Radius ($r$) Diameter ($d$) Circumference ($C$) Area ($A$) Step-by-Step Derivation Circle Equation Calculator Find the standard form of the circle equation, $(x - h)^2 + (y - k)^2 = r^2$, given the center and radius. Standard Form: $(x - h)^2 + (y - k)^2 = r^2$ Center X ($h$) Center Y ($k$) Radius ($r$) Derive Equation Standard Equation General Form Geometric Formulas for Circle Properties The entire geometry of a circle can be calculated if the radius ($r$) is known. The relationships are defined by the constant $\pi \approx 3.14159$ . Property Formula (using $r$) Inverse Formula (to find $r$) Diameter ($d$) $d = 2r$ $r = d/2$ Circumference ($C$) $C = 2\pi r$ $$r = \frac{C}{2\pi}$$ Area ($A$) $A = \pi r^2$ $$r = \sqrt{\frac{A}{\pi}}$$ Circle Equations (Analytic Geometry) In a coordinate plane, the location and size of a circle are described by its equation. The circle is defined as the set of all points $(x, y)$ that are equidistant (distance $r$) from a fixed center point $(h, k)$. Standard Form (Center-Radius Form) This is the simplest form, directly showing the center $(h, k)$ and the radius $r$: $$(x - h)^2 + (y - k)^2 = r^2$$ General Form The general form is found by expanding the standard form: $$x^2 + y^2 + Dx + Ey + F = 0$$ Where the center coordinates are related by $h = -D/2$ and $k = -E/2$. Frequently Asked Questions (FAQ) What is the formula for the Area of a Circle? The area ($A$) of a circle is calculated by multiplying pi ($\pi \approx 3.14159$) by the square of the radius ($r$): $A = \pi r^2$. What is the formula for the Circumference of a Circle? The circumference ($C$), which is the distance around the circle, is calculated as pi ($\pi$) times the diameter ($d$), or two times pi times the radius ($r$): $C = \pi d = 2\pi r$. What is the standard form equation of a circle? The standard form equation of a circle with center $(h, k)$ and radius $r$ is: $(x - h)^2 + (y - k)^2 = r^2$. What is the relationship between the circle and the ellipse? A circle is a special case of an ellipse where the two semi-axes are equal ($a = b = r$) and the eccentricity ($e$) is zero. Geometrically, it is the only conic section with $A=C$ in the general form $A x^2 + C y^2 + D x + E y + F = 0$. Essential Circle Formulas Area $$A = \pi r^2$$ Circumference $$C = 2\pi r$$ Standard Equation $$(x-h)^2 + (y-k)^2 = r^2$$ Related Geometry Tools Ellipse Calculator Area Calculator (General) Arc Length Calculator Sector Area Calculator Distance Formula Calculator

Math & Conversions Cone Volume Calculator Cone Volume Calculator ($\mathbf{V = \frac{1}{3}\pi r^2 h}$) Calculate the volume ($V$), base area ($B$), lateral area ($L$), and slant height ($l$) of a right circular cone. Input the radius ($r$) and the perpendicular height ($h$) below. Enter Dimensions Radius ($r$) Height ($h$, perpendicular) Calculate Properties Key Geometric Results Slant Height ($l$) Base Area ($B$) Lateral Area ($L$) Total Surface Area ($A$) Cone Volume ($V$) Step-by-Step Solution Formulas Used in Cone Geometry All calculations for a right circular cone derive from three fundamental geometric formulas: 1. Cone Volume ($V$) The volume of a cone is precisely one-third the volume of a cylinder with the same base and height. $$V = \frac{1}{3} B h = \frac{1}{3} \pi r^2 h$$ Where $r$ is the radius of the base and $h$ is the perpendicular height. 2. Surface Area Formulas The total surface area ($A$) is the sum of the circular base area ($B$) and the lateral (side) area ($L$). **Base Area ($B$):** Area of the circular base. $$\mathbf{B} = \pi r^2$$ **Lateral Area ($L$):** Area of the slanted side. $$\mathbf{L} = \pi r l$$ **Total Surface Area ($A$):** $$\mathbf{A} = \pi r^2 + \pi r l$$ 3. Slant Height ($l$) The slant height is necessary for surface area calculations. In a right cone, the radius ($r$), perpendicular height ($h$), and slant height ($l$) form a right triangle. Therefore, we use the Pythagorean theorem: $$l = \sqrt{r^2 + h^2}$$ Right Cone vs. Oblique Cone This calculator is designed for a **right cone**, where the apex (tip) is vertically aligned over the center of the base, making the height ($h$) perpendicular to the base radius ($r$). An **oblique cone** has its apex off-center, resulting in a slant height that is not uniform around the circumference. While the volume formula $V = \frac{1}{3} \pi r^2 h$ remains valid if the true perpendicular height ($h$) is known, calculating the surface area of an oblique cone is much more complex. Frequently Asked Questions (FAQ) What is the formula for the volume of a cone? The volume of a cone is one-third of the area of the base times the perpendicular height. The formula is $V = \frac{1}{3} \pi r^2 h$, where $r$ is the radius and $h$ is the height. What is the difference between height (h) and slant height (l)? The **height (h)** is the perpendicular distance from the apex (tip) of the cone to the center of the base. The **slant height (l)** is the distance from the apex to any point on the circumference of the base. In a right cone, they are related by the Pythagorean theorem: $r^2 + h^2 = l^2$. How do I find the base area of a cone? The base of a cone is a circle. Therefore, the base area (B) is calculated using the standard circle area formula: $B = \pi r^2$, where $r$ is the radius. What is the formula for the total surface area of a cone? The total surface area ($A$) is the sum of the base area ($B$) and the lateral surface area ($L$): $A = B + L$. This translates to the formula $A = \pi r^2 + \pi r l$, where $l$ is the slant height. Key Cone Formulas Volume $$V = \frac{1}{3} \pi r^2 h$$ Slant Height $$l = \sqrt{r^2 + h^2}$$ Total Surface Area $$A = \pi r (r + l)$$ Related Geometry Tools Cylinder Volume Calculator Sphere Volume Calculator Solid Geometry Calculator (All Shapes) Area Calculator (Base) Pythagorean Theorem Calculator

Math & Conversions Correlation Calculator Correlation Coefficient Calculator (Pearson's $r$) Use this calculator to find the strength and direction of the linear relationship between two variables, X and Y. Enter your paired data points below. Each line should contain the X-value, followed by the Y-value, separated by a comma or space. Data Set X Data Set Y Make sure X and Y have the same number of data points. Use commas, spaces, or new lines to separate values. Calculate Correlation Results Pearson's $r$ Interpretation Coefficient of Determination ($R^2$) Step-by-Step Calculation Table Pearson's Correlation Coefficient ($r$) Formula Pearson's $r$ is calculated using the following formula, which is designed to measure the covariance of X and Y normalized by the standard deviations of X and Y: $$r = \frac{n \sum xy - (\sum x)(\sum y)}{\sqrt{[n \sum x^2 - (\sum x)^2][n \sum y^2 - (\sum y)^2]}}$$ Where: $n$ is the number of paired observations (data points). $\sum x$ and $\sum y$ are the sums of the X and Y values. $\sum x^2$ and $\sum y^2$ are the sums of the squared X and squared Y values. $\sum xy$ is the sum of the products of X and Y for each pair. This formula always yields a value between $-1$ and $+1$. Interpreting the Correlation Coefficient ($r$) The value of $r$ tells you two things about the relationship between X and Y: Direction (Sign): **Positive ($r > 0$):** As X increases, Y tends to increase (a positive slope). **Negative ($r < 0$):** As X increases, Y tends to decrease (a negative slope). **Zero ($r \approx 0$):** No linear relationship exists. Strength (Magnitude): The closer the value is to $\pm 1$, the stronger the linear relationship. The closer it is to 0, the weaker the relationship. Strength Guidelines (Commonly Used) Magnitude of $|r|$ Strength 0.90 to 1.00 Very Strong 0.70 to 0.89 Strong 0.50 to 0.69 Moderate 0.30 to 0.49 Weak 0.00 to 0.29 Negligible Frequently Asked Questions (FAQ) What does a correlation coefficient measure? The correlation coefficient, typically Pearson's $r$, measures the strength and direction of the linear relationship between two quantitative variables. It is a value between -1 and +1. What is the difference between correlation and causation? Correlation indicates that two variables move together (e.g., ice cream sales and sunscreen sales both rise in summer). Causation means one variable directly causes a change in the other. Correlation does not imply causation (e.g., ice cream sales do not cause sunscreen sales to rise; the common cause is the heat). What is the Coefficient of Determination ($R^2$)? The Coefficient of Determination ($R^2$) is the square of the correlation coefficient ($r^2$). It represents the proportion of the variance in the dependent variable that is predictable from the independent variable. For example, if $r = 0.8$, $R^2 = 0.64$, meaning 64% of the variability in Y can be explained by the relationship with X. When should I use a different correlation method? Pearson's $r$ is best for data that is normally distributed and linear. If your relationship is monotonic (consistently increasing or decreasing but not necessarily linear), Spearman's rank correlation might be more appropriate. If your data involves binary or categorical variables, other measures (like Cramer's V or phi coefficient) are needed. Correlation Interpretation $r = +1.0$: Perfect Positive $r = -1.0$: Perfect Negative $r = 0.0$: No Linear Relationship $r \approx 0.8$: Strong Positive $r \approx -0.4$: Weak Negative Always consider the magnitude and the sign. Related Statistics Tools Correlation Coefficient (Alternative) Linear Regression Calculator Mean, Median, Mode Calculator Standard Deviation Calculator P-Value Calculator t-Test Calculator Covariance Calculator

Math & Conversions Decimal Calculator Decimal Calculator: Rounding & Conversions This comprehensive tool helps you manipulate decimal numbers by performing essential functions: **rounding**, **conversion to fraction**, and **conversion to percentage**. Enter your number and select your rounding preference below. Decimal Number Round to (Decimal Places) 0 (Nearest Whole Number) 1 (Tenths) 2 (Hundredths) 3 (Thousandths) 4 (Ten-Thousandths) 5 (Hundred-Thousandths) Calculate Results Rounded Value As a Fraction As a Percentage Step-by-Step Rounding Understanding Decimals and the Rounding Rule A decimal number is used to represent a non-whole number, where the digits to the right of the decimal point represent fractions with denominators of 10, 100, 1000, and so on (tenths, hundredths, thousandths). The Standard Rounding Rule (Half-Up) The universal rule for rounding is often called "round half up." To round a number to a specific decimal place: Identify the Target Digit: This is the last digit you want to keep. Look Right: Look at the digit immediately to its right (the next digit). Apply the Rule: If the next digit is $\mathbf{5}$ or greater, increase the target digit by one (round up). If the next digit is $\mathbf{4}$ or less, keep the target digit as it is (round down or 'floor'). Example: Round 4.736 to the nearest hundredth (2 decimal places). The target digit is 3. The next digit is 6. Since 6 is $\geq 5$, you round the 3 up to 4. The result is 4.74. Decimal Conversions Decimal to Percentage Converting a decimal to a percentage is straightforward: you multiply the decimal by 100 and add the percent symbol (%). This is mathematically equivalent to moving the decimal point two places to the right. $\text{Percentage} = \text{Decimal} \times 100$ Example: $0.25 = 0.25 \times 100 = 25\%$ Decimal to Fraction To convert a decimal to a fraction, the key is to determine the place value of the last digit. This determines the denominator (e.g., tenths = 10, hundredths = 100, thousandths = 1000). Write the non-whole number part as the numerator. Write the corresponding power of ten as the denominator. Simplify the fraction using the Greatest Common Divisor (GCD). Example: $0.125$. The last digit (5) is in the thousandths place. So, $\frac{125}{1000}$. Dividing both by the GCD (125) gives $\frac{1}{8}$. Frequently Asked Questions (FAQ) What is the rule for rounding decimals? The standard rule is to look at the digit immediately to the right of the place you want to round to. If that digit is 5 or greater (5, 6, 7, 8, or 9), you round up the digit in the target position. If that digit is 4 or less (0, 1, 2, 3, or 4), you keep the digit in the target position the same (round down or 'floor'). How do I convert a decimal to a fraction? 1. Write the decimal as a numerator over a denominator of 1. (e.g., $0.75

Math & Conversions Decimal to Fraction Converter Decimal to Fraction Converter Convert any terminating decimal (e.g., 0.125, 3.4) into its simplest fractional form. Enter your decimal number below to get the results as a **simplified fraction**, an **improper fraction**, and a **mixed number**. Enter Decimal Number Convert to Fraction Conversion Results Simplified Fraction (Lowest Terms) Improper Fraction Mixed Number Step-by-Step Simplification The Three Steps to Converting Decimals to Fractions The key principle is that the place value of the last digit in the decimal determines the denominator (a power of ten) of the initial fraction. Determine Initial Fraction ($A/B$): Identify the place value of the last digit in the decimal. If the decimal is 0.7 (tenths place), the denominator is 10. Fraction: $\frac{7}{10}$ If the decimal is 0.75 (hundredths place), the denominator is 100. Fraction: $\frac{75}{100}$ If the decimal is 0.125 (thousandths place), the denominator is 1000. Fraction: $\frac{125}{1000}$ The whole number part is temporarily ignored or used to convert to an improper fraction later. Find the Greatest Common Divisor (GCD): The GCD is the largest number that divides evenly into both the numerator and the denominator. This is the crucial step for simplification. Simplify: Divide both the numerator and the denominator by the GCD to get the simplified fraction (lowest terms). Handling Decimals with a Whole Number For a number like $4.3$, treat the whole number (4) and the decimal part (0.3) separately. $0.3 = \frac{3}{10}$. The mixed number is $4\frac{3}{10}$. To find the improper fraction, convert the mixed number: $4\frac{3}{10} = \frac{(4 \times 10) + 3}{10} = \frac{43}{10}$. Converting Repeating Decimals This calculator primarily targets terminating decimals. Converting repeating decimals (like $0.\overline{6}$) requires a different algebraic technique: Let $x$ equal the repeating decimal (e.g., $x = 0.666...$). Multiply $x$ by $10^n$, where $n$ is the number of repeating digits (e.g., $10x = 6.666...$). Subtract the original equation from the new one ($10x - x = 6.666... - 0.666...$). Solve for $x$: $9x = 6$, so $x = \frac{6}{9}$, which simplifies to $\frac{2}{3}$. Frequently Asked Questions (FAQ) What is the fastest way to convert a decimal to a fraction? The fastest method is to identify the place value of the last digit. If $0.125$ ends in the thousandths place, the fraction is initially $125/1000$. If $0.8$ ends in the tenths place, the fraction is $8/10$. Then, simplify. How do you convert a decimal like 0.333... (repeating) to a fraction? For a simple repeating decimal like $0.\overline{3}$, the fraction is the repeating digit over 9. So $0.\overline{3} = 3/9$, which simplifies to $1/3$. For $0.\overline{12}$, the fraction is $12/99$. The rule is to use a denominator of 9s (one 9 for each repeating digit). How do I convert a decimal that includes a whole number (e.g., 5.4) to a fraction? For 5.4, the whole number (5) becomes the integer part of the mixed number. The decimal part (0.4) converts to a fraction ($4/10$), which simplifies to $2/5$. The result is $5 \frac{2}{5}$. What is the Greatest Common Divisor (GCD) in fraction conversion? The GCD is the largest number that divides exactly into both the numerator and the denominator of the fraction. Finding the GCD is necessary to ensure the fraction is fully simplified (in its lowest terms). Decimal Place Values 0.1 = Tenths $$\frac{1}{10}$$ 0.01 = Hundredths $$\frac{1}{100}$$ 0.001 = Thousandths $$\frac{1}{1000}$$ Related Math Tools Fraction Calculator Decimal Calculator Mixed Number Converter Percentage Calculator GCD/GCF Calculator Basic Arithmetic

Math & Conversions Distance Calculator Distance Calculator (Euclidean Coordinate Distance) Calculate the shortest straight-line distance ($D$) between two points, $P_1$ and $P_2$, in a two-dimensional (2D) or three-dimensional (3D) coordinate system. Enter the coordinates of both points below. Endpoints $P_1$ and $P_2$ Point X-Coordinate Y-Coordinate Z-Coordinate (Optional) P₁ P₂ Leave Z coordinates blank for a 2D calculation. Calculate Distance Results Distance ($D$) Midpoint ($M$) Step-by-Step Solution The Euclidean Distance Formula The distance formula is a direct application of the Pythagorean Theorem in a coordinate system. It finds the length of the hypotenuse created by the differences in the $x$ and $y$ coordinates ($\Delta x$ and $\Delta y$). Distance Formula (2D) Given points $P_1(x_1, y_1)$ and $P_2(x_2, y_2)$, the distance $D$ is: $$D = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}$$ Distance Formula (3D) For three dimensions, the formula simply adds the difference in the $z$ coordinates: $$D = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2 + (z_2 - z_1)^2}$$ Related Formula: Midpoint The midpoint ($M$) is the coordinate point exactly halfway between the two endpoints. Since the calculation relies on the same input coordinates, this calculator provides it as a helpful related value: $$M = \left(\frac{x_1 + x_2}{2}, \frac{y_1 + y_2}{2}, \dots \right)$$ Frequently Asked Questions (FAQ) What is the Euclidean Distance Formula (2D)? The 2D distance formula finds the length of a segment connecting two points $P_1(x_1, y_1)$ and $P_2(x_2, y_2)$: $D = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}$. How does the distance formula relate to the Pythagorean Theorem? The distance formula is essentially the Pythagorean Theorem ($a^2 + b^2 = c^2$) applied to a coordinate grid. The distance $D$ acts as the hypotenuse ($c$), while the horizontal difference $(\Delta x)$ and vertical difference $(\Delta y)$ act as the legs ($a$ and $b$). In 3D, a third leg $(\Delta z)$ is added. What is the difference between Euclidean and Geodesic distance? Euclidean distance (calculated here) is the straight-line distance in an ideal, flat coordinate system. Geodesic distance is the shortest path between two points on a curved surface (like the surface of the Earth), often calculated using the 'Great Circle Distance' formula for maximum accuracy. Can the distance calculator handle negative coordinates? Yes, the formula correctly handles negative coordinates because the differences are squared, which always results in a positive contribution to the total distance. Core Formulas Distance (2D) $$D = \sqrt{(\Delta x)^2 + (\Delta y)^2}$$ Distance (3D) $$D = \sqrt{(\Delta x)^2 + (\Delta y)^2 + (\Delta z)^2}$$ Midpoint (2D) $$M = \left(\frac{x_1+x_2}{2}, \frac{y_1+y_2}{2}\right)$$ Related Geometry Tools Distance Formula Calculator (Specific) Midpoint Calculator Pythagorean Theorem Calculator Vector Calculator (Magnitude)

Math & Conversions Ellipse Calculator Ellipse Calculator (Conic Sections) Calculate all the geometric and analytical properties of an ellipse. Enter the coefficients from the general form of the conic section: $\mathbf{A x^2 + C y^2 + D x + E y + F = 0}$ where $A$ and $C$ must be positive and not equal. General Form: $A x^2 + C y^2 + D x + E y + F = 0$ A C D E F B ($x y$ term, assumed 0) Calculate Ellipse Properties Key Properties Center $(h, k)$ Standard Form Equation Axes, Foci, and Area Semi-Major Axis ($a$) Semi-Minor Axis ($b$) Foci Distance ($c$) Eccentricity ($e$) Area Approximate Circumference (Ramanujan) Step-by-Step (Completing the Square) The Standard Form of an Ellipse The standard (canonical) form of an ellipse centered at $(h, k)$ is essential for identifying its properties quickly. The equation is: $$\frac{(x-h)^2}{a^2} + \frac{(y-k)^2}{b^2} = 1$$ Where: $(h, k)$: The coordinates of the center. $a$: The semi-major or semi-minor axis associated with the $x$ terms. $b$: The semi-major or semi-minor axis associated with the $y$ terms. The semi-major axis (the larger of $a$ and $b$) determines the orientation of the ellipse (horizontal if under $x$, vertical if under $y$). Finding the Foci and Eccentricity The foci (plural of focus) are two fixed points inside the ellipse used to define its shape. The distance from the center to each focus is denoted by $c$, and it is related to the semi-axes by the Pythagorean-like relationship: $$c^2 = (\text{larger denominator}) - (\text{smaller denominator})$$ $$\text{or } c^2 = a^2 - b^2$$ The **eccentricity** ($e$) measures the flatness of the ellipse and is defined as $e = \frac{c}{a}$ (where $a$ is the length of the semi-major axis). For an ellipse, $0 \le e < 1$. Area and Circumference Unlike a circle, the area calculation for an ellipse is simple, but its circumference is complex. Area: The area enclosed by the ellipse is exactly equal to $\pi$ times the product of the two semi-axes: $$\text{Area} = \pi ab$$ Circumference (Perimeter): The circumference $C$ has no exact algebraic solution. We use the highly accurate approximation formula discovered by Srinivasa Ramanujan: $$C \approx \pi [3(a+b) - \sqrt{(3a+b)(a+3b)}]$$ Frequently Asked Questions (FAQ) What is the relationship between a, b, and c in an ellipse? In an ellipse, '$a$' is always the semi-major axis (the larger of the two), '$b$' is the semi-minor axis (the smaller), and '$c$' is the distance from the center to a focus. They are related by the formula $c^2 = a^2 - b^2$ (or $a^2 = b^2 + c^2$). Note that unlike a hyperbola, '$a$' is *not* always the x-semiaxis. What is the general equation of an ellipse? The most general form is a conic section where $A \neq C$ but share the same sign: $A x^2 + C y^2 + D x + E y + F = 0$. The simpler, standard form (canonical) is $\frac{(x-h)^2}{a^2} + \frac{(y-k)^2}{b^2} = 1$. Is there an exact formula for the circumference of an ellipse? No, the circumference of an ellipse cannot be calculated with a simple, exact algebraic formula. It requires an elliptical integral of the second kind. For practical purposes, approximations are used, such as Ramanujan's approximation, which is highly accurate: $C \approx \pi [3(a+b) - \sqrt{(3a+b)(a+3b)}]$. What is the eccentricity of an ellipse? Eccentricity ($e$) measures how much an ellipse deviates from a perfect circle ($e=0$). It is calculated as $e = c/a$. For an ellipse, $0 \le e < 1$. The higher the eccentricity, the more elongated the ellipse. Key Ellipse Formulas Foci Relationship $$c^2 = a^2 - b^2$$ Area $$Area = \pi ab$$ Eccentricity $$e = \frac{c}{a}$$ Related Conic Section Tools Conic Classifier Hyperbola Calculator Parabola Calculator Circle Calculator Area Calculator (General)

Math & Conversions Heron's Formula Calculator Heron's Formula Calculator (Area from 3 Sides) Heron's Formula is a method to find the area ($\mathcal{A}$) of any triangle when only the three side lengths ($a$, $b$, and $c$) are known. Input the side lengths below to calculate the area and the semi-perimeter ($s$). Enter the Three Side Lengths Side a Side b Side c Calculate Area ($\mathcal{A}$) Key Results Perimeter ($P$) Semi-Perimeter ($s$) Area ($\mathcal{A}$) Step-by-Step Solution Heron's Formula ($\mathcal{A}$) Heron's Formula, attributed to Heron of Alexandria, calculates the area ($\mathcal{A}$) of a triangle using only the lengths of its sides ($a, b, c$). It is expressed in two sequential parts: Part 1: Calculate the Semi-Perimeter ($s$) The semi-perimeter is half the perimeter of the triangle: $$s = \frac{a + b + c}{2}$$ Part 2: Calculate the Area ($\mathcal{A}$) The area is then found by multiplying the semi-perimeter by the difference between the semi-perimeter and each side, and taking the square root of the final product: $$\mathcal{A} = \sqrt{s(s-a)(s-b)(s-c)}$$ The Triangle Inequality Rule Before applying Heron's formula, it is essential to ensure that the sides can actually form a triangle. If the sides do not satisfy the **Triangle Inequality Theorem**, the number inside the square root will be negative, and the area is non-real. The rule requires that the sum of the lengths of any two sides must be greater than the length of the third side: $a + b > c$ $a + c > b$ $b + c > a$ If these conditions are not met, the calculator will indicate an invalid triangle. Frequently Asked Questions (FAQ) What is Heron's Formula used for? Heron's Formula (also called Hero's Formula) is used to find the area of any triangle when the lengths of all three sides are known. It is particularly useful when the height of the triangle is unknown or difficult to determine. What is the semi-perimeter? The semi-perimeter ($s$) is simply half the perimeter of the triangle. It is the intermediate variable required to calculate the area using Heron's Formula: $s = \frac{a + b + c}{2}$. What is the Triangle Inequality Rule? The Triangle Inequality Theorem states that for any triangle to be valid, the sum of the lengths of any two sides must be greater than the length of the third side. For sides a, b, and c, this means: $a+b > c$, $a+c > b$, and $b+c > a$. Does Heron's Formula work for right triangles? Yes, Heron's Formula works for *all* types of triangles, including right triangles, acute triangles, obtuse triangles, and equilateral triangles. For a right triangle, the area can also be found more simply using $A = \frac{1}{2}ab$. Heron's Key Formulas Semi-Perimeter $$s = \frac{a+b+c}{2}$$ Area $$\mathcal{A} = \sqrt{s(s-a)(s-b)(s-c)}$$ Related Triangle Tools Triangle Solver (SSS, SAS, AAS) Area Calculator (General) Perimeter Calculator Law of Cosines Calculator Law of Sines Calculator

Math & Conversions Law of Cosines Calculator Law of Cosines Calculator (Non-Right Triangles) The Law of Cosines is used to solve any general triangle (non-right triangle) when you know either all three sides (**SSS**) or two sides and the angle between them (**SAS**). Select your known case below. Select Your Known Case SSS (Side-Side-Side) SAS (Side-Angle-Side) Side a Side b Side c Angle $\alpha$ ($A$) Angle $\beta$ ($B$) Angle $\gamma$ ($C$) Solve Triangle Solved Triangle Sides Angles Step-by-Step Solution The Three Forms of the Law of Cosines The Law of Cosines is a powerful tool because it works for any triangle. It relates the three sides of a triangle to the cosine of one of its angles. Formulas to Find a Missing Side (SAS Case) If you know two sides and the included angle (SAS), use one of these formulas to find the third side: $$a^2 = b^2 + c^2 - 2bc \cos(\alpha)$$ $$b^2 = a^2 + c^2 - 2ac \cos(\beta)$$ $$c^2 = a^2 + b^2 - 2ab \cos(\gamma)$$ Formulas to Find a Missing Angle (SSS Case) If you know all three sides (SSS), you can rearrange the primary formula to find any angle: $$\alpha = \cos^{-1} \left( \frac{b^2 + c^2 - a^2}{2bc} \right)$$ $$\beta = \cos^{-1} \left( \frac{a^2 + c^2 - b^2}{2ac} \right)$$ $$\gamma = \cos^{-1} \left( \frac{a^2 + b^2 - c^2}{2ab} \right)$$ Law of Cosines vs. Law of Sines The Law of Cosines is typically the starting point for solving a triangle in the **SAS** and **SSS** cases. Once you have used the Law of Cosines to find one missing value, it is often easier and less ambiguous to use the **Law of Sines** to find the remaining angles. The Law of Sines (used in ASA, AAS, and SSA cases) is: $$\frac{a}{\sin(\alpha)} = \frac{b}{\sin(\beta)} = \frac{c}{\sin(\gamma)}$$ The Law of Cosines avoids the "ambiguous case" (SSA), making it the safer choice when applicable. Frequently Asked Questions (FAQ) What is the Law of Cosines used for? The Law of Cosines is a fundamental rule in trigonometry used to find missing parts of any general triangle (non-right triangle). It is specifically used when you know two sides and the included angle (SAS) or all three sides (SSS). What is the Law of Cosines formula for finding side 'a'? The formula to find side 'a' when sides 'b' and 'c' and the included angle 'A' are known is: $a^2 = b^2 + c^2 - 2bc \cos(A)$. This formula is a generalization of the Pythagorean Theorem. When should I use the Law of Cosines instead of the Law of Sines? Use the Law of Cosines in two specific scenarios: \n1. **SSS Case:** When you know all three sides and need to find an angle.\n2. **SAS Case:** When you know two sides and the angle between them (the included angle) and need to find the opposite side. \nFor all other solvable cases (ASA, AAS), the Law of Sines is sufficient. Does the Law of Cosines work for right triangles? Yes. If $\gamma = 90^\circ$, then $\cos(\gamma) = 0$. The formula $c^2 = a^2 + b^2 - 2ab \cos(\gamma)$ simplifies to $c^2 = a^2 + b^2$, which is the Pythagorean Theorem. So, the Law of Cosines is a generalization that always works. Key Cosine Formulas Find Side 'a' (SAS) $$a^2 = b^2 + c^2 - 2bc \cos(\alpha)$$ Find Angle 'A' (SSS) $$\alpha = \cos^{-1} \left( \frac{b^2 + c^2 - a^2}{2bc} \right)$$ Angle Sum Rule $$\alpha + \beta + \gamma = 180^\circ$$ Related Triangle Tools Law of Sines Calculator Triangle Solver (General) Heron's Formula Calculator (Area) Right Triangle Calculator Degrees to Radians Converter

Math & Conversions Law of Sines Calculator Law of Sines Calculator ($\frac{\sin(A)}{a} = \frac{\sin(B)}{b}$) This calculator solves any non-right triangle when provided with two angles and one side (AAS/ASA) or two sides and one non-included angle (SSA). Enter exactly three known values below. The tool will identify the case and provide all solutions. Enter Exactly 3 Known Values Side a Side b Side c Angle $\alpha$ (A) Angle $\beta$ (B) Angle $\gamma$ (C) Solve Triangle Solution 1: Solution 2 (Ambiguous Case): Step-by-Step Solution The Law of Sines Formula The Law of Sines is used to solve oblique triangles (non-right triangles). It establishes a direct proportionality between the side lengths and the sines of their opposite angles: $$\frac{a}{\sin(\alpha)} = \frac{b}{\sin(\beta)} = \frac{c}{\sin(\gamma)}$$ To find a missing value, you must always use a complete ratio (a known side and its opposite angle) as your reference. Triangle Solution Cases for the Law of Sines The Law of Sines is effective in solving three specific input cases, based on the known parts of the triangle: **AAS (Angle-Angle-Side):** Two angles and a non-included side. (Always has one unique solution) **ASA (Angle-Side-Angle):** Two angles and the included side. (Always has one unique solution) **SSA (Side-Side-Angle):** Two sides and a non-included angle. (The **Ambiguous Case**) The Ambiguous Case (SSA) When solving the SSA case, the Law of Sines may yield **zero, one, or two** possible triangles. This ambiguity occurs because the sine function is positive in both the first and second quadrants ($\sin(\theta) = \sin(180^\circ - \theta)$). To determine the number of solutions, the height ($h$) of the triangle is compared to the known side opposite the known angle ($a$) and the adjacent side ($b$). $$h = b \sin(\alpha)$$ **Zero Solutions:** If $a < h$ or $a \le b$ and $\alpha \ge 90^\circ$. **One Solution:** If $a = h$ or $a \ge b$. **Two Solutions:** If $h < a < b$. Frequently Asked Questions (FAQ) What is the Law of Sines formula? The Law of Sines is a ratio relating the length of a side of a triangle to the sine of its opposite angle: $\\frac{a}{\\sin(\\alpha)} = \\frac{b}{\\sin(\\beta)} = \\frac{c}{\\sin(\\gamma)}$. What is the Ambiguous Case (SSA)? The Ambiguous Case (Side-Side-Angle) occurs when the known parts can create zero, one, or two different triangles. This ambiguity arises because the sine of an angle is equal to the sine of its supplement ($\sin(\\theta) = \\sin(180^\\circ - \\theta)$). How do you find the area using the Law of Sines? If you know two sides and the included angle (SAS), the area is calculated by: $\\mathcal{A} = \\frac{1}{2} ab \\sin(\\gamma)$. The Law of Sines is integral to finding the necessary angle for this formula if the full SAS set is not known initially. When should I use the Law of Cosines instead? You should use the Law of Cosines for the SSS (Side-Side-Side) and SAS (Side-Angle-Side) cases, as it is less prone to ambiguity than the Law of Sines in these scenarios. Law of Sines Formulas The Core Ratio $$\frac{a}{\sin(\alpha)} = \frac{b}{\sin(\beta)}$$ Find Angle $\beta$ $$\beta = \sin^{-1}\left(\frac{b \sin(\alpha)}{a}\right)$$ Ambiguous Case (h) $$h = b \sin(\alpha)$$ Related Triangle Tools Law of Cosines Calculator (SSS, SAS) Triangle Solver (General) Heron's Formula Calculator (Area) Right Triangle Calculator Degrees to Radians Converter

Math & Conversions Midpoint Calculator Midpoint Calculator (2D & 3D) Find the exact midpoint $M$ of a line segment connecting two endpoints, $P_1$ and $P_2$. The midpoint is calculated by finding the average of the coordinates for each dimension ($x, y$, and optionally $z$). Endpoints $P_1(x_1, y_1, z_1)$ and $P_2(x_2, y_2, z_2)$ Point X-Coordinate Y-Coordinate Z-Coordinate (Optional) P₁ P₂ Calculate Midpoint Results Midpoint Coordinates (M) Segment Length (Distance) Step-by-Step Calculation The Midpoint Formula (2D and 3D) The midpoint formula is a direct application of the arithmetic mean. For each dimension, the midpoint coordinate is simply the average of the two endpoints' coordinates. Midpoint Formula (2 Dimensions) Given endpoints $P_1(x_1, y_1)$ and $P_2(x_2, y_2)$, the midpoint $M$ is: $$M = \left(\frac{x_1 + x_2}{2}, \frac{y_1 + y_2}{2}\right)$$ Midpoint Formula (3 Dimensions) For three dimensions, the formula is extended to include the $z$ coordinate: $$M = \left(\frac{x_1 + x_2}{2}, \frac{y_1 + y_2}{2}, \frac{z_1 + z_2}{2}\right)$$ The calculation principle remains identical: find the average position along each axis. Related Formula: Distance Between Two Points While the midpoint finds the location of the center of the segment, the distance formula finds the length of the segment ($D$). The 2D distance formula is: $$D = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}$$ Frequently Asked Questions (FAQ) What is the midpoint formula? The midpoint formula is the method used to find the coordinates of the point that lies exactly halfway between two given points, $P_1(x_1, y_1)$ and $P_2(x_2, y_2)$. The formula is $M = \left(\frac{x_1 + x_2}{2}, \frac{y_1 + y_2}{2}\right)$. Is the midpoint formula just the average? Yes, the midpoint formula is essentially finding the arithmetic mean (average) of the x-coordinates and the y-coordinates separately. Since the midpoint is equidistant from both endpoints, finding the average position gives you the exact middle point. Can the midpoint formula be used in 3D (three dimensions)? Yes. In three dimensions, you simply extend the formula to include the z-coordinate: $M = \left(\frac{x_1 + x_2}{2}, \frac{y_1 + y_2}{2}, \frac{z_1 + z_2}{2}\right)$ How do I find a missing endpoint if I know the midpoint? You can rearrange the formula to solve for a missing coordinate. For the x-coordinate, the formula is: $x_2 = 2M_x - x_1$. This works because the midpoint ($M_x$) is halfway between $x_1$ and $x_2$. Core Geometry Formulas Midpoint (2D) $$M = \left(\frac{x_1 + x_2}{2}, \frac{y_1 + y_2}{2}\right)$$ Distance (2D) $$D = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}$$ Missing Endpoint ($x_2$) $$x_2 = 2 M_x - x_1$$ Related Coordinate Geometry Tools Distance Formula Calculator Slope Calculator Coordinate Calculator Intercept Calculator Vector Calculator

Math & Conversions Prism Calculator Prism Calculator (Volume and Surface Area) Calculate the volume ($V$), lateral surface area ($LSA$), and total surface area ($TSA$) for the two most common types of prisms: **Rectangular** and **Triangular**. Select the prism type below to input your dimensions. Rectangular Prism (Cuboid) Triangular Prism Base is a Rectangle: $V = l \cdot w \cdot h$ Length (l) Width (w) Height of Prism (h) Base is a Triangle: $V = \frac{1}{2} b \cdot h_{base} \cdot L$ Triangular Base Dimensions: Base Side ($b$) Base Height ($h_{base}$) Prism Length/Height ($L$) Base Side Lengths (for Area Calculation): Side 1 ($s_1$) Side 2 ($s_2$) Side 3 is the Base Side ($b$). For TSA, enter the three sides of the triangular base. Calculate Prism Results Lateral Surface Area ($LSA$) Total Surface Area ($TSA$) Volume ($V$) Step-by-Step Solution Universal Prism Formulas A prism is a polyhedron whose two bases are parallel, congruent polygons, and whose faces are parallelograms. All calculations stem from the base properties. 1. Volume ($V$) The volume formula is consistent for all prisms, requiring only the area of the base ($B$) and the height/length of the prism ($L$): $$V = B \cdot L$$ Where $B$ is the area of the polygonal base and $L$ is the distance between the two bases. 2. Surface Area ($TSA$ and $LSA$) The lateral surface area ($LSA$) is the area of all the sides, excluding the top and bottom bases. The total surface area ($TSA$) includes the bases. Lateral Surface Area ($LSA$): $$LSA = P \cdot L \quad (\text{where } P \text{ is the perimeter of the base})$$ Total Surface Area ($TSA$): $$TSA = LSA + 2B = P \cdot L + 2B$$ Specific Prism Formulas Rectangular Prism (Cuboid) For a rectangular base with length $l$ and width $w$, and prism height $h$: **Base Area ($B$):** $B = l \cdot w$ **Base Perimeter ($P$):** $P = 2l + 2w$ **Volume ($V$):** $V = l \cdot w \cdot h$ **Total Area ($TSA$):** $TSA = 2(lw + lh + wh)$ Triangular Prism For a triangular base with base side $b_{tri}$ and height $h_{tri}$, and prism length $L$: **Base Area ($B$):** $B = \frac{1}{2} b_{tri} \cdot h_{tri}$ **Volume ($V$):** $V = \frac{1}{2} b_{tri} \cdot h_{tri} \cdot L$ **Total Area ($TSA$):** You must sum the area of the three rectangular sides (Lateral Area) and the two triangular bases. Frequently Asked Questions (FAQ) What is the universal formula for the volume of any prism? The volume of any prism, regardless of the shape of its base, is found by multiplying the area of the base ($B$) by the height ($L$) of the prism: $V = B \cdot L$. What is the difference between total surface area and lateral surface area? The **lateral surface area** ($LSA$) is the area of all the sides, excluding the top and bottom bases. The **total surface area** ($TSA$) includes the area of the two bases plus the lateral surface area: $TSA = LSA + 2B$. How do I find the volume of a rectangular prism? A rectangular prism (or cuboid) has a rectangular base. The base area ($B$) is length times width ($l \cdot w$). Therefore, the volume is $V = l \cdot w \cdot h$, where $h$ is the prism height. What is a right prism versus an oblique prism? A **right prism** has lateral faces that are rectangles, meaning the lateral edges are perpendicular (at $90^\circ$) to the base. An **oblique prism** has lateral faces that are parallelograms and are not perpendicular to the base. The volume formula $V=B \cdot L$ only works for the oblique prism if $L$ is the *perpendicular* distance between the bases. Universal Prism Formulas Volume $$V = B \cdot L$$ Total Surface Area $$TSA = P \cdot L + 2B$$ Base Area (Rectangle) $$B = l \cdot w$$ Related Solid Geometry Tools Rectangular Prism Volume Pyramid Calculator Cylinder Volume Calculator Sphere Volume Calculator Volume Calculator (General)

Math & Conversions Regular N-gon Calculator Regular N-gon Calculator (Area, Apothem, Radius) Solve for all geometric properties of any regular polygon. Input the **Number of Sides ($n$)** and **one** known dimension (Side Length $s$, Apothem $a$, or Radius $R$). Input (Minimum $n$ and one dimension) Number of Sides ($n$) Side Length ($s$) Apothem ($a$) Radius ($R$) Calculate Polygon Properties Key Results Perimeter ($P$) Apothem ($a$) Radius ($R$) Area ($A$) Angle Measures (Degrees) Internal Angle ($\theta_i$) Central Angle ($\theta_c$) Exterior Angle ($\theta_e$) Step-by-Step Solution The Fundamental Triangle and Trigonometry The key to solving any regular N-gon is to divide it into $n$ congruent **isosceles triangles**. Each triangle is formed by two radii ($R$) and one side ($s$). Bisecting this triangle creates a **right triangle**, known as the fundamental triangle, with the following properties: **Leg 1:** Apothem ($a$). **Leg 2:** Half of the side length ($s/2$). **Hypotenuse:** Radius ($R$). **Angle at Center:** Half of the central angle ($\frac{180^\circ}{n}$). Using this right triangle and the tangent function, we establish the core relationship: $$\tan\left(\frac{180^\circ}{n}\right) = \frac{\text{Opposite}}{\text{Adjacent}} = \frac{s/2}{a}$$ Key Formulas for a Regular Polygon Once one property (like the apothem) is known, all other properties can be calculated. Property Formula Perimeter ($P$) $P = n \cdot s$ Area ($A$) $$A = \frac{1}{2} a P \quad \text{or} \quad A = \frac{1}{4} n s^2 \cot\left(\frac{180^\circ}{n}\right)$$ Internal Angle ($\theta_i$) $$\theta_i = \frac{(n-2) \times 180^\circ}{n}$$ Central Angle ($\theta_c$) $$\theta_c = \frac{360^\circ}{n}$$ Frequently Asked Questions (FAQ) What is a regular N-gon? A regular N-gon (regular polygon) is a closed, two-dimensional shape with 'N' number of equal sides and 'N' equal interior angles. Examples include the square (N=4), pentagon (N=5), and hexagon (N=6). What is the apothem? The apothem ($a$) is the distance from the center of a regular polygon to the midpoint of any side. It is the radius of the inscribed circle and is used as the height when calculating the area of the polygon's fundamental triangle. What is the relationship between the central angle and the number of sides? The central angle ($\theta_c$) is the angle formed by two radii drawn to consecutive vertices. It is calculated by dividing the full circle ($360^\circ$) by the number of sides ($n$): $\theta_c = 360^\circ

Math & Conversions Regular Polygon Calculator Regular Polygon Calculator (Area, Perimeter, Apothem, Angles) Solve for all geometric properties of any regular polygon. Input the **Number of Sides ($n$)** and **one** known dimension (Side Length $s$, Apothem $a$, or Radius $R$). Input (Minimum $n$ and one dimension) Number of Sides ($n$) Side Length ($s$) Apothem ($a$) Radius ($R$) Calculate Polygon Properties Key Results Perimeter ($P$) Apothem ($a$) Radius ($R$) Area ($A$) Angle Measures (Degrees) Internal Angle ($\theta_i$) Central Angle ($\theta_c$) Exterior Angle ($\theta_e$) Angle Sums (Total) Sum of Internal Angles Sum of Exterior Angles Step-by-Step Solution The Fundamental Triangle and Trigonometry The key to solving any regular N-gon is to divide it into $n$ congruent **isosceles triangles**. Each triangle is formed by two radii ($R$) and one side ($s$). Bisecting this triangle creates a **right triangle**, known as the fundamental triangle, with the following properties: **Leg 1:** Apothem ($a$). **Leg 2:** Half of the side length ($s/2$). **Hypotenuse:** Radius ($R$). **Angle at Center:** Half of the central angle ($\frac{180^\circ}{n}$). Using this right triangle and the tangent function, we establish the core relationship: $$\tan\left(\frac{180^\circ}{n}\right) = \frac{\text{Opposite}}{\text{Adjacent}} = \frac{s/2}{a}$$ Key Formulas for a Regular Polygon Once one property (like the apothem) is known, all other properties can be calculated. Property Formula Perimeter ($P$) $P = n \cdot s$ Area ($A$) $$A = \frac{1}{2} a P \quad \text{or} \quad A = \frac{1}{4} n s^2 \cot\left(\frac{180^\circ}{n}\right)$$ Internal Angle ($\theta_i$) $$\theta_i = \frac{(n-2) \times 180^\circ}{n}$$ Central Angle ($\theta_c$) $$\theta_c = \frac{360^\circ}{n}$$ Frequently Asked Questions (FAQ) What defines a regular polygon? A regular polygon (or N-gon) is a closed, two-dimensional shape with 'N' number of equal sides and 'N' equal interior angles. Examples include the square (N=4), pentagon (N=5), and hexagon (N=6). What is the formula for the sum of the interior angles? The sum of all interior angles of any N-sided polygon (regular or irregular) is found using the formula: $\\text{Sum} = (n - 2) \times 180^\\circ$. What is the difference between Apothem and Radius? The **Radius ($R$)** is the distance from the center to a vertex, forming the circumscribed circle. The **Apothem ($a$)** is the perpendicular distance from the center to the midpoint of a side, forming the inscribed circle. What is the formula for the area using only the side length ($s$)? The area can be calculated using only the side length ($s$) and the number of sides ($n$) with the cotangent function: $\\text{Area} = \frac{1}{4} n s^2 \cot(\\frac{180^\\circ}{n})$. Key Polygon Formulas Area $$A = \frac{1}{2} a P$$ Internal Angle $$\theta_i = \frac{(n-2)180^\circ}{n}$$ Apothem (from Side $s$) $$a = \frac{s}{2 \tan\left(\frac{180^\circ}{n}\right)}$$ Related Geometry Tools Regular N-gon Calculator (Alternate) Polygon Interior Angles Area Calculator (General) Perimeter Calculator Irregular Polygon Calculator

Math & Conversions Right Triangle Calculator Right Triangle Calculator (Sides, Angles, Area) This tool solves for all unknown values of a right-angled triangle. Enter the minimum two known values (either two sides, or one side and one acute angle) in the fields below. The angle $\gamma$ is assumed to be $90^\circ$ (opposite side $c$). Input Known Dimensions ($\gamma = 90^\circ$) Side a (Leg) Side b (Leg) Side c (Hypotenuse) Angle $\alpha$ (degrees) Angle $\beta$ (degrees) Clear All Solve Right Triangle Solved Properties Sides & Hypotenuse Angles ($\alpha, \beta$) Area ($A$) Perimeter ($P$) Step-by-Step Solution Fundamental Formulas for Right Triangles Solving a right triangle relies on the Pythagorean theorem and the basic trigonometric identities (SOHCAHTOA). 1. The Pythagorean Theorem (Sides) This is the relationship between the lengths of the three sides ($a, b$, legs; $c$, hypotenuse): $$a^2 + b^2 = c^2$$ 2. Trigonometric Ratios (Angles) Used to find the angles when side lengths are known (using inverse functions, $\sin^{-1}$, $\cos^{-1}$, $\tan^{-1}$): $$\sin(\alpha) = \frac{\text{Opposite } a}{\text{Hypotenuse } c} \quad \cos(\alpha) = \frac{\text{Adjacent } b}{\text{Hypotenuse } c} \quad \tan(\alpha) = \frac{\text{Opposite } a}{\text{Adjacent } b}$$ 3. Area and Perimeter **Area ($A$):** Since the legs $a$ and $b$ are perpendicular, they serve as the base and height: $$A = \frac{1}{2} a b$$ **Perimeter ($P$):** The sum of all three sides: $$P = a + b + c$$ Frequently Asked Questions (FAQ) What is the Pythagorean Theorem? The Pythagorean Theorem states that in a right-angled triangle, the square of the hypotenuse (the side opposite the right angle, $c$) is equal to the sum of the squares of the other two sides (legs, $a$ and $b$): $a^2 + b^2 = c^2$. What is SOHCAHTOA? SOHCAHTOA is a mnemonic device used to remember the three basic trigonometric ratios: SOH: $\sin(\theta) = \frac{\text{Opposite}}{\text{Hypotenuse}}$ CAH: $\cos(\theta) = \frac{\text{Adjacent}}{\text{Hypotenuse}}$ TOA: $\tan(\theta) = \frac{\text{Opposite}}{\text{Adjacent}}$ What is a Pythagorean Triple? A Pythagorean Triple is a set of three positive integers $a, b,$ and $c$ such that $a^2 + b^2 = c^2$. The most famous example is $(3, 4, 5)$. What is the area of a right triangle? The area ($A$) of a right triangle is half the product of the two legs (the sides that form the right angle, $a$ and $b$): $A = \frac{1}{2} a b$. Triangle Diagram Angle $\alpha$ is opposite side $a$. Angle $\beta$ is opposite side $b$. $\gamma = 90^\circ$ is opposite hypotenuse $c$. Key Right Triangle Formulas Side Relationship $$a^2 + b^2 = c^2$$ Angle Sum Rule $$\alpha + \beta = 90^\circ$$ Area $$A = \frac{1}{2} ab$$ Related Geometry Tools Pythagorean Theorem Calculator Angle Calculator (General) Triangle Calculator (Law of Sines/Cosines) Distance Calculator

Math & Conversions Sphere Volume Calculator Sphere Volume Calculator ($V = \frac{4}{3} \pi r^3$) Calculate the volume ($V$), surface area ($A$), and circumference ($C$) of a sphere. Enter either the **Radius ($r$)** or the **Diameter ($d$)** below, and the tool will solve for all other properties. Enter EITHER Radius ($r$) or Diameter ($d$) Radius ($r$) Diameter ($d$) Calculate Sphere Key Results Radius ($r$) Surface Area ($A$) Circumference ($C$) Volume ($V$) Step-by-Step Solution Formulas for Sphere Volume and Area The volume and surface area formulas for a sphere, first rigorously derived by Archimedes, demonstrate a simple and elegant relationship with the radius, $r$. 1. Volume ($V$) The volume of the sphere is determined by cubing the radius: $$V = \frac{4}{3} \pi r^3$$ Historically, Archimedes showed that the volume of a sphere is two-thirds the volume of the smallest cylinder that can enclose it. 2. Surface Area ($A$) The surface area formula is exactly four times the area of the sphere's great circle ($\pi r^2$, the cross-section passing through the center): $$A = 4 \pi r^2$$ Key Sphere Terminology **Radius ($r$):** The distance from the center to any point on the surface. **Diameter ($d$):** The distance across the sphere passing through the center ($d = 2r$). **Great Circle:** The largest circle that can be drawn on the surface of the sphere. Its circumference is $C = 2\pi r$. Frequently Asked Questions (FAQ) What is the formula for the volume of a sphere? The volume ($V$) of a sphere is given by the formula: $V = \frac{4}{3} \pi r^3$, where $r$ is the radius. What is the formula for the surface area of a sphere? The surface area ($A$) of a sphere is given by the formula: $A = 4 \pi r^2$, which is four times the area of its great circle. How does a sphere's volume relate to a cylinder's volume? The volume of a sphere is $\frac{2}{3}$ the volume of a cylinder that perfectly encloses it (a circumscribed cylinder). This relationship was a source of great pride for Archimedes, who first derived these formulas. Is a sphere a polyhedron? No. A polyhedron is a three-dimensional shape with flat polygonal faces (like a cube or pyramid). A sphere has a continuously curved surface, making it a curved geometric shape, not a polyhedron. Sphere Key Formulas Volume $$V = \frac{4}{3} \pi r^3$$ Surface Area $$A = 4 \pi r^2$$ Circumference $$C = 2 \pi r$$ Related Solid Geometry Tools Cylinder Volume Calculator Cone Volume Calculator Volume Calculator (General) Surface Area Calculator (General) Area Calculator (Circle)

Math & Conversions Tangent Line Calculator Tangent Line Calculator Find the equation of the tangent line to the function $f(x)$ at the specified point $x_0$. The tangent line is a linear approximation of the curve at that exact point. Equation: $f(x)$ and Point $x_0$ Function $f(x)$ Point $x_0$ Calculate Tangent Line Tangent Line Equation ($y = mx + b$) Slope ($m = f'(x_0)$) Point of Tangency $(x_0, y_0)$ Step-by-Step Calculation The Three Essential Steps to Finding the Tangent Line The equation of the tangent line is found by applying the point-slope formula, $y - y_0 = m(x - x_0)$. This requires three key components derived from calculus: 1. Find the Slope ($m$) The slope is the value of the first derivative of the function $f'(x)$ evaluated at the point $x_0$: $$m = f'(x_0)$$ 2. Find the Point of Tangency ($x_0, y_0$) The $y$-coordinate ($y_0$) is found by substituting $x_0$ into the original function: $$y_0 = f(x_0)$$ 3. Construct the Equation Substitute $m$, $x_0$, and $y_0$ into the point-slope form and algebraically rearrange it into the common slope-intercept form, $y = mx + b$. Tangent Line vs. Normal Line The **Normal Line** is perpendicular to the tangent line at the point of tangency. This relationship is defined by their slopes: If the slope of the tangent line is $m$, the slope of the normal line ($m_n$) is the negative reciprocal: $$m_n = -\frac{1}{m}$$ The normal line equation is often used in optics (reflection/refraction) and fluid dynamics. Frequently Asked Questions (FAQ) What is the relationship between the derivative and the tangent line? The derivative of a function $f(x)$ at a specific point $x_0$ is mathematically equal to the slope of the tangent line to the graph of $f(x)$ at that point. $m = f'(x_0)$. What is the formula for the tangent line? The most fundamental formula is the point-slope form: $y - y_0 = m(x - x_0)$, where $m = f'(x_0)$ and $y_0 = f(x_0)$. What is the normal line? The normal line is the line that is perpendicular to the tangent line at the point of tangency. Its slope is the negative reciprocal of the tangent line's slope: $m_{normal} = -1

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Table of Contents 1. Introduction 2. Information We Collect 3. How We Use Information 4. Data Storage & Security 5. Cookies & Tracking 6. Third-Party Services 7. Your Rights 8. Children's Privacy 9. International Transfers 10. Policy Changes 11. Contact Us 1. Introduction Welcome to CalcDomain. This Privacy Policy explains how Yeah Up S.r.l. ("we," "us," or "our"), operating CalcDomain.com, collects, uses, discloses, and safeguards your information when you visit our website and use our calculators. We are committed to protecting your privacy and being transparent about our data practices. This policy complies with: The European Union General Data Protection Regulation (GDPR) The California Consumer Privacy Act (CCPA) Other applicable data protection laws Key Principle: CalcDomain operates on a privacy-first model. We do not store your calculation data, track your activity for advertising purposes, or sell your personal information to third parties. 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Data Storage & Security 4.1 Where Your Data is Processed Calculator computations occur entirely in your browser . We do not transmit, store, or process your calculator inputs or results on our servers. Data you voluntarily provide (contact forms, newsletter subscriptions) is stored on secure servers within the European Union, in compliance with GDPR requirements. 4.2 Security Measures We implement industry-standard security measures: HTTPS encryption for all data transmission Secure server configurations with regular security updates Access controls limiting who can view personal data Regular security audits and vulnerability assessments Encrypted database storage Automatic logout for admin sessions 4.3 Data Retention We retain personal information only as long as necessary: Contact Form Submissions: 2 years from last contact Newsletter Subscriptions: Until you unsubscribe Log Files: 30 days Analytics Data: 14 months (anonymized) After retention periods expire, data is permanently deleted from our systems. 5. Cookies & Tracking Technologies 5.1 Essential Cookies We use minimal, essential cookies required for website functionality: Session cookies: Maintain your session while browsing (expires when you close browser) Cookie consent: Remember your cookie preferences (expires after 12 months) 5.2 Analytics Cookies (Optional) With your consent, we use Google Analytics with IP anonymization to understand how our site is used. You can opt out at any time. 5.3 What We DON'T Use We do NOT use: Advertising cookies Social media tracking pixels Cross-site tracking Fingerprinting technologies Third-party advertising networks 5.4 Managing Cookies You can control cookies through your browser settings. Note that disabling essential cookies may affect website functionality. Instructions for popular browsers: Chrome: Settings → Privacy and security → Cookies Firefox: Settings → Privacy & Security → Cookies Safari: Preferences → Privacy → Cookies Edge: Settings → Privacy, search, and services → Cookies 6. Third-Party Services 6.1 Service Providers We Use We work with carefully selected service providers: Google Analytics (Optional) Purpose: Website usage analytics Data Shared: Anonymized usage data only Privacy Policy: Google Privacy Policy Opt-out: Google Analytics Opt-out CDN & Hosting Purpose: Fast, reliable website delivery Data Shared: Technical data required for content delivery Location: EU-based servers 6.2 Data Sharing We do NOT sell, rent, or trade your personal information to third parties for marketing purposes. We may share data only in these limited circumstances: Legal Requirements: When required by law, court order, or government request Service Providers: With vendors who help us operate the website (under strict confidentiality agreements) Business Transfers: In case of merger, acquisition, or sale (with notice to users) Protection: To protect rights, property, or safety of CalcDomain, users, or others 7. Your Privacy Rights 7.1 GDPR Rights (EU/EEA Users) If you're in the European Union or European Economic Area, you have the right to: Access: Request a copy of personal data we hold about you Rectification: Correct inaccurate or incomplete data Erasure ("Right to be Forgotten"): Request deletion of your personal data Restriction: Limit how we use your data Data Portability: Receive your data in machine-readable format Object: Object to processing based on legitimate interests Withdraw Consent: Withdraw consent at any time (e.g., unsubscribe from emails) Lodge Complaint: File a complaint with your local data protection authority 7.2 CCPA Rights (California Residents) California residents have the right to: Know: What personal information we collect and how we use it Delete: Request deletion of your personal information Opt-Out: Opt out of sale of personal information (Note: We don't sell data) Non-Discrimination: Equal service regardless of privacy rights exercise 7.3 How to Exercise Your Rights To exercise any of these rights: Email us at: privacy@calcdomain.com Include "Privacy Rights Request" in the subject line Provide sufficient detail for us to verify your identity Specify which right(s) you wish to exercise We will respond within 30 days (GDPR) or 45 days (CCPA) of receiving your request. 8. Children's Privacy CalcDomain is not directed to children under the age of 13 (or 16 in the EU). We do not knowingly collect personal information from children. If you are a parent or guardian and believe your child has provided us with personal information, please contact us at privacy@calcdomain.com . We will promptly delete such information. Our calculators are educational tools that can be used by all ages, but we do not collect personal information from users regardless of age. 9. International Data Transfers CalcDomain is operated by Yeah Up S.r.l., an Italian company. Our servers are located in the European Union, ensuring compliance with GDPR standards. If you access our website from outside the EU, your information may be transferred to, stored, and processed in the EU. By using CalcDomain, you consent to this transfer. We ensure appropriate safeguards are in place for international transfers, including standard contractual clauses approved by the European Commission. 10. Changes to This Privacy Policy We may update this Privacy Policy from time to time to reflect changes in our practices, technology, legal requirements, or other factors. When we make changes: We will update the "Last Updated" date at the top of this page Significant changes will be prominently posted on our website For material changes, we may notify users via email (if we have your email address) We encourage you to review this Privacy Policy periodically. Continued use of CalcDomain after changes indicates acceptance of the updated policy. 11. Contact Us If you have questions, concerns, or requests regarding this Privacy Policy or our data practices, please contact us: Data Protection Contact: Yeah Up S.r.l. Privacy Team Email: privacy@calcdomain.com Website: CalcDomain.com/contact EU Data Protection Authority: If you're not satisfied with our response, you have the right to lodge a complaint with your local supervisory authority. Response Time: We strive to respond to all privacy inquiries within 5 business days for initial acknowledgment and within 30 days for complete resolution. Privacy Policy Summary ✅ What We DO: • Process calculations in your browser • Use minimal analytics to improve service • Protect your data with encryption • Comply with GDPR and CCPA • Respond to your privacy requests ❌ What We DON'T DO: • Store your calculation data • Sell your information • Use advertising trackers • Share data for marketing • Track you across websites

Tool developed by Ugo Candido , a strategic consultant specializing in finance and insurance. Content reviewed by the CompareTheMarket Expert Team . Last reviewed for accuracy on: October 15, 2023.

Tool developed by Ugo Candido . Content reviewed by HealthCalc Expert Team. Last reviewed for accuracy on: October 15, 2023.

Tool developed by Ugo Candido . Content reviewed by the Finance Expert Team. Last reviewed for accuracy on: October 1, 2023.

Tool developed by Ugo Candido . Content reviewed by the MathCalc Expert Team . Last reviewed for accuracy on: October 1, 2023.

Welcome to the Scientific Calculator. This tool is designed for students, engineers, and professionals who need to perform complex mathematical calculations with precision and ease. Enter Expression * Calculate Results Result 0 Data Source and Methodology All calculations are based on standard mathematical operations and principles. For comprehensive understanding, consult reliable mathematical resources. The Formula Explained Example Formula: Result = Expression Glossary of Terms Expression: A combination of numbers, operators, and variables that represent a mathematical operation. Result: The outcome of the mathematical calculation. How It Works: A Step-by-Step Example Enter a mathematical expression, such as 2 + 3 * 4 . The calculator will follow the order of operations, multiplying 3 and 4 first, then adding the result to 2, yielding a final result of 14. Frequently Asked Questions (FAQ) What is a scientific calculator? A scientific calculator is a type of electronic calculator that is designed to perform complex mathematical calculations. How do I use the scientific calculator? Enter your mathematical expression in the input field and press 'Calculate' to see the result. Can the calculator handle trigonometric functions? Yes, the calculator can evaluate functions like sin, cos, and tan using standard JavaScript syntax. Is the calculator accurate? The calculator uses JavaScript's built-in functions for accuracy, but it's always best to double-check complex calculations. What other mathematical operations are supported? In addition to basic arithmetic, the calculator can handle exponentiation, square roots, and more. Tool developed by Ugo Candido . Content reviewed by the Example Inc. Expert Team . Last reviewed for accuracy on: October 15, 2023.

What is a frequency distribution? A frequency distribution is a summary of how often different values occur within a dataset. How do I interpret frequency distribution results? Interpreting frequency distribution involves analyzing the data to find patterns, such as the mode, median, or range. Can frequency distributions be used for continuous data? Yes, frequency distributions can be applied to both discrete and continuous data by using class intervals. What are the benefits of using a frequency distribution calculator? A frequency distribution calculator simplifies and speeds up the process of organizing and analyzing data. How accurate are the results from this calculator? The calculator provides highly accurate results based on standard mathematical methods for frequency distributions.

What is CalcDomain? CalcDomain is a free online platform providing 50+ professional calculators for financial planning, health metrics, business calculations, and more. We're operated by Yeah Up S.r.l., an Italian company established in 2018, and our calculators are built and reviewed by a multidisciplinary team of engineers, MBAs, physicians, and certified professionals.