Pythagoras Theorem

Have you ever wondered how to find a shortcut across a park without measuring it? Imagine a park with a path along two sides, like the edges of a big square corner. But you could walk the shortcut, cutting straight across the corner!

That corner shape is called a right triangle, which just means it has one perfect square corner. The two short sides are the paths you already know. The long side, the shortcut, is called the hypotenuse, which just means the longest side.

Now here is the magic trick Pythagoras found about two thousand five hundred years ago. Draw a square on each side of the triangle and count the little boxes inside. The blue square has nine little boxes inside it.

• Pythagoras, born ~570 BC

Now let us count the boxes in the green square on the bottom side. The green square has sixteen little boxes. Now add them together: nine plus sixteen equals twenty five little boxes total.

If you draw a square on the shortcut side, it will have exactly twenty five boxes! So the shortcut side must be five, because five times five is twenty five. Here is the rule written as a sum: short side squared, plus short side squared, equals long side squared.

A common mistake is adding the sides first and then squaring, but you must square each side first! This is really useful in real life, like when a ladder leans against a wall. You know the wall is four metres and the floor gap is three metres, so the ladder must be five metres long.

• Square FIRST, then add!

This saves you from guessing and buying a ladder that is too short! Now your turn: a triangle has short sides of three and four. What is the long side? Remember: square the two short sides, add them, and that gives you the square of the shortcut side.

• No more guessing ladder length!