# Right Triangle Calculator

Solve right triangles from two sides or a side and angle. Find all sides, angles, area, and perimeter instantly. Free online right triangle calculator.

## What this calculates

Solve any right triangle by entering two known sides or one side and an acute angle. This calculator finds all three sides, both acute angles, the area, and the perimeter using the Pythagorean theorem and trigonometric functions.

## Inputs

- **Known Values** — options: Two Sides, Side and Angle — Choose whether you know two sides or a side and an acute angle.
- **Side a (opposite angle A)** — min 0 — One leg of the right triangle.
- **Side b (opposite angle B)** — min 0 — The other leg of the right triangle.
- **Hypotenuse (c)** — min 0 — The longest side, opposite the right angle. Use instead of one of the legs.
- **Acute Angle (degrees)** — min 0, max 90 — One of the acute angles (for side-and-angle method). This is angle A, opposite side a.

## Outputs

- **Side a** — The leg opposite angle A.
- **Side b** — The leg opposite angle B.
- **Hypotenuse (c)** — The side opposite the right angle.
- **Angle A** — The acute angle opposite side a.
- **Angle B** — The acute angle opposite side b.
- **Area** — The area of the right triangle.
- **Perimeter** — The sum of all three sides.

## Details

A right triangle has one 90-degree angle and two acute angles that sum to 90 degrees. The side opposite the right angle is the hypotenuse (always the longest side), and the other two sides are called legs.

Key Formulas

- Pythagorean Theorem: a^2 + b^2 = c^2 (c is the hypotenuse)

- Trigonometric Ratios: sin(A) = opposite/hypotenuse = a/c, cos(A) = adjacent/hypotenuse = b/c, tan(A) = opposite/adjacent = a/b

- Area: A = 0.5 x leg1 x leg2

- Perimeter: P = a + b + c

Right triangles are the foundation of trigonometry and appear constantly in construction (roofs, ramps, stairs), navigation (distance and bearing calculations), physics (force decomposition), and computer graphics (rotation and projection). Common right triangles include the 3-4-5, 5-12-13, and 8-15-17 Pythagorean triples, as well as the 45-45-90 and 30-60-90 special triangles.

## Frequently Asked Questions

**Q: How do I find the hypotenuse of a right triangle?**

A: Use the Pythagorean theorem: c = sqrt(a^2 + b^2), where a and b are the two legs. For example, if the legs are 3 and 4, the hypotenuse is sqrt(9 + 16) = sqrt(25) = 5. If you know one leg and an angle, use c = a / sin(A) or c = b / cos(A).

**Q: What is a Pythagorean triple?**

A: A Pythagorean triple is a set of three positive integers (a, b, c) where a^2 + b^2 = c^2. Common examples include (3, 4, 5), (5, 12, 13), (8, 15, 17), and (7, 24, 25). Any multiple of a triple is also a triple, so (6, 8, 10) and (9, 12, 15) are also valid.

**Q: What are the special right triangles?**

A: The two special right triangles are the 45-45-90 triangle (isosceles, with legs in ratio 1:1:sqrt(2)) and the 30-60-90 triangle (with sides in ratio 1:sqrt(3):2). These triangles have exact trigonometric values and appear frequently in geometry and standardized tests.

**Q: How do I find an angle in a right triangle from two sides?**

A: Use inverse trigonometric functions. If you know the opposite side and hypotenuse: angle = arcsin(opposite/hypotenuse). If you know the adjacent and hypotenuse: angle = arccos(adjacent/hypotenuse). If you know opposite and adjacent: angle = arctan(opposite/adjacent).

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Source: https://vastcalc.com/calculators/math/right-triangle
Category: Math
Last updated: 2026-04-21