The Pythagorean Theorem: 5 Elegant Proofs Explained

Explore five visual, algebraic, and geometric proofs of a² + b² = c², from ancient Babylon to Einstein.

The Most Famous Theorem in Geometry

The relationship between the legs (a, b) and hypotenuse (c) of a right-angled triangle forms the bedrock of trigonometry, navigation, and modern physics.

Proof 1: Geometric Rearrangement

Arrange four identical right triangles with sides a, b, c inside a square of side length (a + b). The uncovered inner area is a tilted square of area c². Rearranging the four triangles forms two squares of areas a² and b².

Proof 2: Algebraic Expansion

The total area of the large outer square (a + b)² equals the sum of the four triangle areas plus the inner square c².

Proof 3: Similar Triangles (Einstein's Method)

Dropping an altitude from the right angle to the hypotenuse divides the original triangle into two smaller, similar right triangles.

Real-World Applications

From GPS 3D trilateration algorithms to structural civil engineering and video game collision physics, right triangle math executes billions of times every second.