The Pythagorean theorem states that in any right-angled triangle, the square of the hypotenuse (the longest side, opposite the right angle) equals the sum of the squares of the other two sides. Written as a² + b² = c², it is one of the most fundamental relationships in geometry and has practical applications in architecture, navigation, construction, computer graphics, and physics. This page explains the theorem, its three algebraic forms, and multiple real-world worked examples.