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.
A 2,500-year-old equation, three letters long, and we still use it to lay foundations, calibrate GPS, and frame video games. Most theorems retire. This one just keeps working.
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.
Computing screen diagonals, GPS distance, raytracing in graphics, structural diagonals in carpentry, even the 'as the crow flies' shortcut on a city grid — all live or die on Pythagoras.
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.
A ladder is 5 m long and rests against a wall, with its base 3 m from the wall. How high up does it reach?
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.
Pythagoras only works for right triangles. If the triangle has no 90° corner, you need the law of cosines instead. Check the right angle is there before squaring sides.
- a² + b² = c² for right triangles. c is the hypotenuse.
- Distance between two points is just Pythagoras with horizontal and vertical gaps.
- Memorise a few triples (3-4-5, 5-12-13) and you'll spot them everywhere.