Pythagorean Theorem

In any right triangle, the square of the hypotenuse equals the sum of the squares of the other two sides.

The three squares — area of c² equals area of a² plus area of b²

Definition

In any right triangle — a triangle with one $90°$ corner — there is a special relationship between the three sides. The two shorter sides are called legs, and the longest side (opposite the right angle) is called the hypotenuse.

If you build a square on each side of a right triangle, the area of the square on the hypotenuse exactly equals the combined area of the squares on the two legs. This is the Pythagorean Theorem:

$a^2 + b^2 = c^2$

where $a$ and $b$ are the legs and $c$ is the hypotenuse.

Finding the hypotenuse

A right triangle has legs of length $3$ and $4$ . How long is the hypotenuse?

$c^2 = 3^2 + 4^2 = 9 + 16 = 25$

$c = \sqrt{25} = 5$

The hypotenuse is $5$ units long. The triple $(3, 4, 5)$ is probably the most famous right triangle in the world.

Try it

A right triangle has legs of length $5$ and $12$ . Find the length of the hypotenuse.

Solution

$c^2 = 5^2 + 12^2 = 25 + 144 = 169$

$c = \sqrt{169} = 13$

The hypotenuse is $13$ units. The triple $(5, 12, 13)$ is another classic Pythagorean triple.

Related concepts

Needs first

Triangles

Law of Cosines Similarity

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