CalcDomain

Triangle Calculator

This professional triangle calculator solves any triangle from common input sets (SSS, SAS, ASA/AAS, SSA). It’s designed for students, engineers, and educators who need fast, accurate results with full geometric properties and accessible, mobile-first UX.

Interactive Calculator

Choose solving method
Angles are handled in degrees.
Inputs for SSS (three sides)

Results

Results update after you calculate. The area below is reserved to prevent layout shifts.

Solution Summary

Waiting for inputs

Enter the required values and press Calculate.

Authoritative Data Source and Methodology

Authoritative Source: Weisstein, Eric W. “Triangle.” MathWorld—A Wolfram Web Resource. Last updated 2024. Direct link: https://mathworld.wolfram.com/Triangle.html. Additional reference: NIST Digital Library of Mathematical Functions (DLMF), Release 1.1.10, 2023: https://dlmf.nist.gov/.

Tutti i calcoli si basano rigorosamente sulle formule e sui dati forniti da questa fonte.

The Formulas Explained

$$\textbf{Law of Cosines:}\quad a^2=b^2+c^2-2bc\cos A,\; b^2=a^2+c^2-2ac\cos B,\; c^2=a^2+b^2-2ab\cos C$$ $$\textbf{Law of Sines:}\quad \frac{a}{\sin A}=\frac{b}{\sin B}=\frac{c}{\sin C}$$ $$\textbf{Angle Sum:}\quad A+B+C=180^\circ$$ $$\textbf{Heron’s Formula:}\quad s=\frac{a+b+c}{2},\; \text{Area}=\sqrt{s(s-a)(s-b)(s-c)}$$ $$\textbf{Heights:}\quad h_a=\frac{2\,\text{Area}}{a},\; h_b=\frac{2\,\text{Area}}{b},\; h_c=\frac{2\,\text{Area}}{c}$$ $$\textbf{Medians:}\quad m_a=\tfrac12\sqrt{2b^2+2c^2-a^2}\; \text{(cyclic)}$$ $$\textbf{Inradius and Circumradius:}\quad r=\frac{\text{Area}}{s},\; R=\frac{a}{2\sin A}=\frac{b}{2\sin B}=\frac{c}{2\sin C}$$

Glossary of Variables

How It Works: A Step-by-Step Example

Scenario: SSS with a=7, b=8, c=9.

  1. Use the Law of Cosines to find angles: $$A=\cos^{-1}\!\left(\frac{b^2+c^2-a^2}{2bc}\right),\; B=\cos^{-1}\!\left(\frac{a^2+c^2-b^2}{2ac}\right),\; C=180^\circ-A-B.$$ Numerically, A≈46.57°, B≈56.25°, C≈77.18°.
  2. Compute semiperimeter s=(7+8+9)/2=12. Then area: $$\text{Area}=\sqrt{12(12-7)(12-8)(12-9)}=\sqrt{12\cdot5\cdot4\cdot3}=\sqrt{720}\approx26.833.$$
  3. Perimeter P=7+8+9=24. Heights: $$h_a=\frac{2\cdot 26.833}{7}\approx7.666,\; h_b\approx6.708,\; h_c\approx5.963.$$
  4. Inradius r=Area/s≈26.833/12≈2.236. Circumradius R=a/(2 sin A)≈7/(2·0.725)≈4.827.

Frequently Asked Questions (FAQ)

What input combinations are valid?

Any of SSS, SAS, ASA/AAS always yield a unique triangle. SSA may yield zero, one, or two solutions (ambiguous case).

How does the calculator handle SSA?

It applies the Law of Sines: sin B = (b sin A)/a. If 0 < (b sin A)/a < 1 and a < b, two solutions exist. If equal to 1, one right triangle; if greater than 1, no solution.

Are the formulas based on Euclidean geometry?

Yes. All results assume plane Euclidean geometry and standard trigonometric functions in degrees.

How are rounding and precision handled?

You can choose 0–10 decimal places. Internally, computations maintain full precision and are rounded only for display.

Can I detect triangle type?

Yes. The results classify by sides (equilateral, isosceles, scalene) and by angles (acute, right, obtuse).

What if my inputs violate the triangle inequality?

The tool warns you, highlights the problematic fields, and explains how to adjust values to satisfy a+b>c (and cyclic permutations).

Can I use radians?

This version uses degrees for accessibility and clarity. Convert radians to degrees by multiplying by 180/π.