What is Pythagoras' Theorem?
Imagine you're standing at the corner of a rectangular football pitch. You could walk along one side and then the other to reach the opposite corner — or you could cut straight across diagonally. Which path is shorter? The diagonal, of course! But exactly how long is that diagonal? This is the kind of question that Pythagoras' theorem helps us answer.
Over 2,500 years ago, the Greek mathematician Pythagoras discovered a beautiful relationship between the three sides of a right-angled triangle. This relationship is one of the most famous and useful results in all of mathematics.
In a right-angled triangle, the square of the hypotenuse is equal to the sum of the squares of the other two sides. Written algebraically: , where is the hypotenuse.
The longest side of a right-angled triangle, always found directly opposite the right angle (90°).
Identifying the Parts of a Right-Angled Triangle
Before we can use Pythagoras' theorem, we need to be able to identify the parts of a right-angled triangle correctly.
Every right-angled triangle has:
- One right angle (90°), usually marked with a small square in the corner
- Two shorter sides that form the right angle — these are sometimes called the legs
- One longest side opposite the right angle — this is the hypotenuse
To find the hypotenuse, look for the side that is opposite the right angle. It's always the longest side. If you put your finger on the little square (right angle marker) and look across the triangle, the side facing you is the hypotenuse.
A common mistake is labelling any long-looking side as the hypotenuse. The hypotenuse is always opposite the right angle — its position, not just its appearance, is what defines it.
10 more sections in this topic
Pick this up in your Library: it holds the whole topic, notes, cheatsheet and questions.