Pythagoras’ Theorem tells us that the sides of a right angled triangle are related in the following way:

$(\text{shorter}_1)^2+(\text{shorter}_2)^2=(\text{hypotenuse})^2$

If we begin by labelling our triangle with letters $a, b$ and $c$ as on the following triangle,

we have

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

This is a beautifully simple theorem, however it is not that obvious why it works. It is named after Pythagoras, not because he discovered it but because he was probably the first to explain why it works – or the first to have his explanation recorded.

Pythagoras of Samos lived an intriguing life. Read about him on the Mathigon Timeline.

It’s possible to calculate the length of the hypotenuse without the Pythagorean theorem – check out Short story part 1. The theorem however, really simplifies the process – see short story part 2.

Visual explanation of the Pythagorean Theorem

The diagram on the left is constructed with four congruent triangles (four ‘equal’ triangles) and one square.

The diagram on the right is constructed with the same four congruent triangles and with two smaller squares to fill in the gaps.

Made by Daniel Pearcy.

This illustrates why $c^2$ must be equal to $a^2+b^2$

Sometimes the letters in a formula and the letters in a problem might not match up. For example, the formula $c^2=a^2+b^2$ assumes that we label the hypotenuse $c$ . In practice the hypotenuse might be called $x$ or $y$ or $b$ or fence or anything else for that matter. That is why it is important to remember that it is the shorter sides squared, then added, that give the hypotenuse squared.