# Hypotenuse Calculator

Calculate the hypotenuse of a right triangle with the Pythagorean theorem. Also solve for a missing leg. Shows perimeter, area, and step-by-step formula.

## What this calculates

Find the hypotenuse or a missing leg of any right triangle using the Pythagorean theorem. Enter two known sides and the calculator handles the rest. A classic 3-4-5 right triangle has legs of 3 and 4, giving a hypotenuse of 5.

## Inputs

- **Solve for** — options: Hypotenuse (c), Missing Leg (a or b) — Choose whether to find the hypotenuse or a missing leg.
- **Side 1** — Length of the first known side.
- **Side 2** — If finding hypotenuse: the other leg. If finding a leg: the hypotenuse.

## Outputs

- **Missing Side** — The calculated missing side length.
- **Perimeter** — The sum of all three sides.
- **Area** — Area of the right triangle (0.5 x base x height).
- **Formula Used** — formatted as text — The Pythagorean theorem equation applied.

## Details

**The Pythagorean Theorem**

For any right triangle with legs a and b and hypotenuse c:

**a² + b² = c²**

Rearranged to find the hypotenuse: c = sqrt(a² + b²). To find a missing leg: a = sqrt(c² - b²).

**Example: Finding the Hypotenuse**

If the two legs are 6 and 8:
c = sqrt(6² + 8²) = sqrt(36 + 64) = sqrt(100) = 10

**Example: Finding a Missing Leg**

If the hypotenuse is 13 and one leg is 5:
a = sqrt(13² - 5²) = sqrt(169 - 25) = sqrt(144) = 12

**Common Pythagorean Triples**

Some right triangles have all-integer side lengths. The most well-known:
- 3, 4, 5
- 5, 12, 13
- 8, 15, 17
- 7, 24, 25

Any multiple of these also works (e.g., 6, 8, 10 is just 2 times the 3-4-5 triple).

**Perimeter and Area**

The perimeter of a right triangle is simply a + b + c. The area is (1/2) x a x b, since the two legs serve as base and height.

## Frequently Asked Questions

**Q: What is the hypotenuse?**

A: The hypotenuse is the longest side of a right triangle, located directly opposite the 90-degree angle. It is always longer than either leg. In the formula a² + b² = c², c represents the hypotenuse.

**Q: Can the hypotenuse be shorter than a leg?**

A: No. The hypotenuse is always the longest side of a right triangle. If you enter a hypotenuse value smaller than the given leg, the calculator will flag it as an error because no such triangle can exist.

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

A: A Pythagorean triple is a set of three positive integers (a, b, c) where a² + b² = c². Examples include (3, 4, 5), (5, 12, 13), and (8, 15, 17). Any scalar multiple of a triple is also a triple.

**Q: Does the Pythagorean theorem work for non-right triangles?**

A: No. The Pythagorean theorem only applies to right triangles (those with a 90-degree angle). For other triangles, you need the Law of Cosines: c² = a² + b² - 2ab cos(C).

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Source: https://vastcalc.com/calculators/math/hypotenuse
Category: Math
Last updated: 2026-04-08