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 $c$).

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):

2. Trigonometric Ratios (Angles)

Used to find the angles when side lengths are known (using inverse functions, $\sin^{-1}$, $\cos^{-1}$, $\tan^{-1}$):

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)

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a^2 + b^2 = c^2

\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}

A = \frac{1}{2} a b

P = a + b + c

\alpha + \beta = 90^\circ

$\sin(\alpha) = \frac{\text{Opposite } a}{\text{Hypotenuse } c} \quad \cos(\alpha) = \frac{\text{Adjacent } b}{\text{Hypotenuse } c} \quad \tan(\alpha) = \frac{\text{Opposite } a}{\text{Adjacent } b}$