# Hypotenuse Calculator

Calculate the length of the hypotenuse in a right triangle using two sides or one side and one angle with the calculator below.

Solve Using:

## Hypotenuse:

c =

Learn how we calculated this below

## How to Find the Hypotenuse of a Right Triangle

The hypotenuse is the longer side of a right triangle that is opposite the 90° right angle.

A right triangle is composed of the hypotenuse c, two legs a & b, two angles α & β, and its right angle.

The hypotenuse c is the longest side of the triangle in the illustration below. You can solve the hypotenuse using a little trigonometry and one of the following formulas.

### Formula One: Given Two Legs

The Pythagorean theorem can be used to find the hypotenuse using the length of the other two legs. The Pythagorean theorem states a² + b² = c².

This formula can be rewritten to solve for the hypotenuse c:

c = a² + b²

Thus, the hypotenuse c is equal to the square root of leg a squared plus leg b squared. ### Formula Two: Given One Leg and the Adjacent Angle

Given the length of one leg and the adjacent angle, you can find the hypotenuse using the formula:

c = a / cos(β)

The hypotenuse c is equal to side a divided by the cosine of the adjacent angle β. ### Formula Three: Given One Leg and the Opposite Angle

Given the length of one leg and the opposite angle, you can find the hypotenuse using the Law of Sines with the formula:

c = a / sin(α)

The hypotenuse c is equal to side a divided by the sine of the opposite angle α. ### Formula Four: Given One Leg and the Area

If you know the length of one of the legs and the triangle area, you can find the length of the hypotenuse by using the area formula to solve for the length of the other leg, then using the Pythagorean theorem.

Given the formula to find the area of a right triangle, start by finding the length of the other leg:

A = 1 / 2ab

You can rearrange this formula to solve for leg b like this:

b = 2A / a

The length of leg b is equal to 2 times the area A divided by the length of leg a.

Then, you can use the length of legs a and b to find the hypotenuse using the first formula above:

c = a² + b²

## How to Find the Hypotenuse for a 45 45 90 Right Triangle

A 45 45 90 triangle is a special right triangle that is also an isosceles triangle. As the name implies, a 45 45 90 has two 45° angles and one right angle.

Since this is an isosceles triangle, both legs are equal in length, so you can find the length of the hypotenuse of a 45 45 90 right triangle using a simplified formula derived from the Pythagorean theorem.

c = a√2

The length of c is equal to the length of leg a times the square root of 2. ## How to Find the Hypotenuse for a 30 60 90 Right Triangle

A 30 60 90 triangle is a special right triangle that has one 30° angle, one 60° angle, and one right angle.

In this special case, the length of the hypotenuse is always equal to the length of the short leg a of the triangle. So, given the length of the short leg, the formula to find the hypotenuse is:

c = 2a

The length of c is equal to 2 times the length of a. 