How do you prove it?
Arrange four identical right triangles inside a big square with side a+b, leaving a tilted square of side c in the middle. The big square's area (a+b)² equals the four triangles (2ab) plus the middle square (c²). Simplify and you get a² + b² = c². Over 400 different proofs are known.
Where is it used?
The distance formula between two points, √((x₂−x₁)² + (y₂−y₁)²), is the theorem in disguise. Carpenters use the 3-4-5 rule to check square corners; it's also behind TV screen sizes and GPS distance calculations. It bears Pythagoras's name, but Babylonian clay tablets from around 1800 BCE already list Pythagorean triples.
FAQ
- What's a Pythagorean triple?
- Three whole numbers satisfying a²+b²=c², like (3,4,5), (5,12,13), and (8,15,17).
- What about non-right triangles?
- Use the law of cosines: c² = a² + b² − 2ab cos C.
Sources reviewed: 2026-10-25
References
- Euclid. Elements, Book I, Proposition 47.
- Maor, E. (2007). The Pythagorean Theorem: A 4,000-Year History. Princeton University Press.