The Lesson

The hypotenuse is the longest side in a right triangle (also called a right-angled triangle). It is the side opposite the 90° angle (called a right angle).

Dictionary Definition

The Oxford English Dictionary defines the hypotenuse as "the side of a right-angled triangle which subtends, or is opposite to, the right angle."

The Hypotenuse and Trigonometry

Trigonometry relates an angle within a right triangle to the lengths of its three sides (called the hypotenuse, adjacent and opposite). The image below shows what we mean by an angle (labelled θ) and the three sides (the hypotenuse, adjacent and opposite). The adjacent and opposite sides are defined in relation to the angle (see Note).

  • The sine function relates the angle in a right triangle to the ratio of the length of the opposite side to the length of the hypotenuse:

  • The cosine function relates the angle in a right triangle to the ratio of the length of the adjacent side to the length of the hypotenuse:

The Hypotenuse and Pythagoras' Theorem

Pythagoras' theorem concerns the relationship between the length of the hypotenuse and the lengths of the other two sides. Pythagoras' theorem (or the Pythagorean theorem) states that:
The square of the hypotenuse of a right triangle is equal to the sum of the squares of the other two sides.
It is easier to remember Pythagoras' theorem as a formula:
a2 + b2 = c2
This formula can be rearranged to show the length of the hypotenuse in terms of the other two sides:

In the formula, c is the length of the hypotenuse and a and b are the lengths of the other two sides. The image below shows what we mean:

What's in a Name?

"Hypotenuse" comes from the Greek word "hypoteínousa", which combines "hypo" ("under") and teíno ("I stretch" or "length"). The hypotenuse is the "line under" the right angle.