Hypotenuse: triangle, sine & calculate

In the context of right triangles, you have probably heard about the hypotenuse of the triangle. The hypotenuse is a special side of the triangle.

It is important for you that when you look at a triangle you can quickly see which side of the triangle is the hypotenuse or whether a triangle has a hypotenuse at all.

This article aims to clear up all of your potential question marks about the hypotenuse of a triangle. You will also learn two ways to calculate the hypotenuse.

Calculate trigonometry hypotenuse

The hypotenuse is a designation for a triangle side specifically in the right triangle. So if a triangle doesn’t have a right angle, then it doesn’t have a hypotenuse either!

triangle hypotenuse

In a right triangle, the sides of the triangle have special names.

The other two sides of the triangle are called catheti. They are said to abut or enclose the right angle.

In this right triangle ABC, side c is the hypotenuse of the triangle because it is opposite the right angle. The sides a and b that touch the right angle are the legs of the triangle.

Figure 1: Right triangle ABC with hypotenuse c

Hypotenuse as the longest side in the triangle

You may have heard that the hypotenuse is the longest side of a triangle. But why is it like that?

In a triangle, the longest side is opposite the largest angle. Because of the sum of interior angles in a triangle we also know that all three angles add up to 180°. This means that the right angle of 90° is the largest of the three angles in a triangle (if a second angle were greater than 90°, the sum of these two angles would be greater than 180°).

If the argument with the sum of the interior angles is not entirely clear to you, read our article «Sum of the interior angles of a triangle» again.

It now follows that the side opposite the right angle is the longest side in the triangle. And that is exactly our hypotenuse according to the definition!

Figure 1: Graphic illustrating the hypotenuse as the longest side of a triangle

You can rotate any right triangle like the triangle above.

This representation shows directly that side b – the hypotenuse – is longer than sides a and c. Why?

The semicircle is created by drawing a circle with radius b around point C. The distance s thus indicates how much longer the triangle side b is than the triangle side a.

This works analogously for the circle, the circle around point A with radius b. Here you can see from the segment t that side b is longer than side c.

Hypotenuse formula – Pythagorean theorem

Depending on the given sizes of the triangle, there are several ways to calculate the length of the hypotenuse or to calculate other sizes (lengths or angles) of the triangle given the hypotenuse.

One of these methods is the calculation using the Pythagorean theorem.

Pythagorean theorem basic knowledge

As a reminder, the formulation of the Pythagorean theorem again:

If the right angle is not opposite side c, the variables in the formula must be adjusted accordingly. For example, in a triangle with, the formula .

Figure 3: right triangle with adjusted Pythagoras formula (right angle at point B)

Calculation with the Pythagorean theorem

Given the two legs a and b of the right triangle, the length of the hypotenuse can be calculated using Pythagoras:

sine hypotenuse

In many cases, however, only one leg of the right triangle is given. If, in addition, the size of an interior angle that differs from a right angle (often also referred to as an acute interior angle) is given, the length of the hypotenuse can be calculated with sine and cosine.

Sine and cosine basic knowledge

Sines and cosines indicate length ratios in a right-angled triangle. It can be precisely defined as follows:

The side adjacent to is that of the two sides that is in contact with the angle.

Figure 4: Adjacent and opposite leg of an angle

For example:

If you’re not sure what the sine and cosine of a right triangle mean, read the Sine and cosine of a right triangle article again.

Calculation with sine and cosine

The formulas for sine and cosine can be rearranged to calculate the hypotenuse.

Out of

and

follows by rearranging the terms:

Depending on which angle and which leg is given, the appropriate of the two formulas must be selected.