Hypotenuse

The longest side of a right triangle is called the hypotenuse. It lies opposite the right angle. The other two sides of a triangle go by a variety of names. Generally, they are referred to as the legs of the triangle, but they are also referred to as the opposite side or adjacent side and also the base or altitude of the triangle.

What is the hypotenuse

The hypotenuse is one of the sides in a right triangle that lies opposite the right angle of the triangle. It is the longest side of a right triangle. Note that the longest side of any triangle that is not a right triangle is not a hypotenuse. Only right triangles have a hypotenuse. The figure below shows a right triangle example and a hypotenuse example.

In this figure, the bottom leg may be referred to as the base while the left leg may be referred to as the altitude. Given the angle θ, the bottom leg would be referred to as the adjacent side while the left leg would be referred to as the opposite side.

It makes sense that the hypotenuse lies opposite the right angle of the triangle since the length of the sides 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

To find the hypotenuse of a triangle, there are a few different methods depending on the available information.

Hypotenuse formula

The hypotenuse formula refers to the Pythagorean Theorem when it is solved for the hypotenuse, c. The Pythagorean Theorem 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, the Pythagorean Theorem states that:

Solving for c gives us the hypotenuse formula:

Below is a hypotenuse example using the Pythagorean Theorem.

Using law of sines

To find the hypotenuse using the law of sines, reference the following figure and use the formula:

Thus, to find the hypotenuse using the law of sines, we need to know the measure of one angle and one leg. If we do, finding the hypotenuse is just a matter of plugging values into the formula.

Triangle trigonometric relationships

It is also possible to use triangle trigonometry with sine, cosine, and tangent to find the hypotenuse given a side and an angle of the triangle. 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.

Using area and one leg

To find the hypotenuse using the area and one leg, use the formula:

where b is the base of the triangle.