# Hypotenuse

The **hypotenuse** of a right triangle (or right-angled triangle) is the side opposite its right angle. It is the longest side of a right triangle.

We can find the hypotenuse of a triangle using the Pythagorean Theorem if we know the length of the two legs (*a* and *b*). The formula is:

Download this **calculator** to get the results of the formulas on this page. Choose the initial data and enter it in the upper left box. For results, press ENTER.

Triangle-total.rar or Triangle-total.exe

Note. Courtesy of the author: **José María Pareja Marcano**. Chemist. Seville, Spain.

## Exercise

Find the ** hypotenuse** of the triangle shown below:

We apply the equation:

So the hypotenuse is **5 cm**.

## Relationship between Legs and Hypotenuse

### Pythagorean Theorem

The **Pythagorean Theorem**, also known as Pythagoras’s Theorem, is a fundamental relation in Euclidean Geometry among the three sides of a right triangle (2 legs and hypotenuse). This theorem can be written as the following equation:

### Geometric Mean Theorem

The **Geometric Mean Theorem** (or Altitude-on-Hypotenuse Theorem) relates the height (*h*) of the triangle and the legs of two triangles similar to the main *ABC*, by plotting the height *h* over the hypotenuse, stating that in every right triangle, the height (*h*) relative to the hypotenuse is the geometric mean of the two projections of the legs on the hypotenuse (*n* and *m*).

### Leg Rule

The leg rule is a theorem that relates the segments projected by the legs on the hypotenuse with the legs they touch.

In every right triangle, a leg (*a* or *b*) is the geometric mean between the hypotenuse (*c*) and the projection of that leg on it (*n* or *m*).