The formula
a² + b² = c²
Where it comes from
In a right triangle, the square built on the hypotenuse has exactly the same area as the two squares on the legs combined. That is the theorem, and it is a statement about areas before it is a formula about lengths.
How to work it out by hand
- Identify the hypotenuse: it is the side opposite the right angle, always the longest
- To find it, square both legs and add them
- To find a leg, square the hypotenuse and subtract the known leg squared
- Take the square root
What is worth knowing
The theorem was known in practice long before Pythagoras: Babylonian tablets list right-triangle triples a thousand years earlier, and Egyptian rope-stretchers used the 3-4-5 triangle to lay out right angles. What the Greeks contributed was the proof. It also has a converse that is genuinely useful on site: if a² + b² = c², the triangle *must* be right-angled, which is why the 3-4-5 trick squares a corner reliably. And it generalises — the law of cosines is the same statement with a correction term for triangles that are not right-angled.