# Hypotenuse

The side opposite the right angle of a right triangle is called the hypotenuse. The sides that form the right angle are called legs, or sometimes the adjacent or opposite side (relative to one of the angles of the triangle that is not the right angle), depending on the context.

The length of a side of a triangle corresponds to the size of the angle opposite the side. Since the hypotenuse of a right triangle is the longest side of the triangle, the 90° angle opposite it is also the largest angle of the right triangle. This also makes sense because the internal angles of a triangle sum to 180°. Since all triangles have 3 sides and 3 internal angles, it is impossible for a right triangle to have another angle that is greater than or equal to 90°, because the third angle would have to be 0° or have a negative angle measurement.

## How to find the hypotenuse of a right triangle

The hypotenuse of a right triangle can be found using the Pythagorean Theorem, which is a theorem specific to right triangles. It cannot be used with non-right triangles. For a right triangle with a hypotenuse of length c and leg lengths a and b:

or

Example:

Find the hypotenuse length of the triangle below.

Given legs a = 15 and b = 20:

c^{2} = 15^{2} + 20^{2}

c^{2} = 625

c = 25

So, the hypotenuse length is 25.

It is also possible to find the hypotenuse of a triangle given a side and an angle of the triangle, however this requires the use of trigonometry. Refer to the trigonometry section for more detail. Briefly, given the following right triangle

the following relationships can be used to find the various sides and angles of a right triangle:

sin(A) = ; sin(B) =

cos(A) = ; cos(B) =

tan(A) = ; tan(B) =

In the relationships above, A, B, and C are the angles of the triangle opposite the sides a, b, and c, respectively.