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 obvious. It is named after Pythagoras, not because he discovered it but because he was probably the first to prove it.

Here is a visual, showing a square with side length $c$ , a square with side length $a$ and a square with side length $b$ .

Made by Daniel Pearcy.

Without using any algebra, this illustrates why $c^2$ must be equal to $a^2+b^2$ . Again, not obvious to begin with, so keep looking to see why.

Explanation

One difficulty with using letters in any formula is that the letters in a formula and the letters in a problem might not match up. For example, the formula above assumes that we label the hypotenuse $c$ , but in any problem 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.