Pythagorean Theorem at a Glance

A long time ago, when philosophy ruled and Socrates drank hemlock, a brainiac named Pythagoras proved that for right triangles:

a² + b² = c²

Subsequent admirers and right-angle folks know this equation as the Pythagorean Theorem. Sides a and b are the legs, and side c is the hypotenuse (the one opposite the right angle).

This means that if we know the length of two sides of a right triangle, we have the third side in the bag. Just remember how to use your squares and square roots and you're good to go.

Sample Problem

In the following triangle, how long is side c?

First, jot down the Pythagorean Theorem.

a² + b² = c²

We're looking for the hypotenuse, which means we know a and b. Next, plug in 4 for a and 3 for b.

4² + 3² = c²

Keep going.

16 + 9 = c²
c² = 25

Take the square root of both sides. (We're rooting for you.)

c = 5

There we go. Missing side = found.

Sample Problem

What's the length of side a in this triangle?

Rearrange the Pythagorean Theorem to solve for a².

a² + b² = c²

a² = c² – b²

Plug in b = 6 and c = 10.

a² = 10² – 6²
a² = 100 – 36
a² = 64

Take the square root of both sides for the finishing touch.

a = 8

Sample Problem

How long is side b in this triangle?

Arrange the Pythagorean Theorem in terms of b².

a² + b² = c²

b² = c² – a²

Plug 'em in, then solve for b with some clever square-rooting.

b² = 13² – 12²
b² = 169 – 144
b² = 25
b = 5