General | 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