Geometry
Pythagoras
Drag the legs of a right triangle and watch the squares on each side. The two smaller ones always add up to the big one.
The orange and blue squares (a² and b²) together always equal the green square on the hypotenuse (c²). That is Pythagoras.
In a right-angled triangle, the square on the longest side equals the sum of the squares on the other two. It's the most useful theorem in elementary maths — distance, navigation, screens, and graphics all use it.
c is the hypotenuse — the side opposite the right angle.
Legs 3 and 4 — find the hypotenuse.
3² + 4² = 9 + 16 = 25. √25 = 5. So c = 5.
Hypotenuse 13, one leg 5 — find the other.
5² + b² = 13² → b² = 169 − 25 = 144 → b = 12.
Distance between two points
On a grid, the distance from (x₁, y₁) to (x₂, y₂) is √((x₂−x₁)² + (y₂−y₁)²) — Pythagoras with sides Δx and Δy.
Triples like 3-4-5, 5-12-13, 8-15-17 always make a right triangle. Builders still use the 3-4-5 trick to square corners.