The Pythagorean theorem – basics and applications
Right triangles, Pythagorean triples and practical use
The Pythagorean theorem (c² = a² + b²) is one of the best-known theorems in geometry. It applies exclusively to right triangles: the hypotenuse c (the side opposite the right angle) is always the longest side. The legs a and b form the right angle. Conversely: if three side lengths satisfy the formula a² + b² = c², the triangle necessarily has a right angle.
The best-known Pythagorean triple is (3, 4, 5): 3² + 4² = 9 + 16 = 25 = 5². The ancient Egyptians already used this triple with a rope with 12 knots (3 + 4 + 5 = 12) to mark right angles when building the pyramids. Other integer triples: (5, 12, 13), (8, 15, 17), (7, 24, 25). These triples remain useful in trades and construction to this day.
In computer graphics and game development, the Pythagorean theorem is everywhere: the Euclidean distance between two points (x₁, y₁) and (x₂, y₂) is calculated as d = √((x₂−x₁)² + (y₂−y₁)²). This distance is used for collision detection, pathfinding and rendering. In three dimensions, this becomes d = √((x₂−x₁)² + (y₂−y₁)² + (z₂−z₁)²) by applying the theorem twice.
Our Pythagorean theorem calculator covers all three use cases: calculating the hypotenuse c from legs a and b, leg a from b and c, or leg b from a and c. Enter the two known sides and select the value you want to find. The calculator returns the missing side as well as all three side lengths accurate to six decimal places.