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 ($\gamma = 90^\circ$)
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):
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: $$P = a + b + c$$