Geometry

Pythagorean Theorem: Proof and Applications

By SC Editorial 2026-06-11 8 min read

The Pythagorean theorem states that in a right triangle, a^2 + b^2 = c^2, where c is the hypotenuse. It is one of the most famous results in mathematics, with over 400 known proofs.

The Formula

Given legs a and b and hypotenuse c: c = sqrt(a^2 + b^2). Try our Pythagorean Theorem Calculator for instant results.

Visual Proof Sketch

Arrange four copies of a right triangle around a square. The area of the large square equals the sum of areas of two smaller squares built on the legs - demonstrating a^2 + b^2 = c^2.

Applications

Construction workers use it to ensure corners are square. Navigation systems compute distances. Computer graphics rely on it for vector magnitudes.

Pythagorean Triples

Integer solutions include (3,4,5), (5,12,13), and (8,15,17). Recognizing triples speeds up mental math.

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