Pythagorean Theorem Calculator
Solve any right triangle using the Pythagorean theorem. Enter any two sides to find the third — including hypotenuse, adjacent, or opposite sides.
Pythagorean Theorem Guide
The Theorem
For any right triangle: a² + b² = c², where c is the hypotenuse (the side opposite the right angle, always the longest side). Named after Pythagoras of Samos (570–495 BC), though the relationship was known to Babylonian and Indian mathematicians centuries earlier. The theorem applies only to right triangles — for other triangles, use the Law of Cosines: c² = a² + b² − 2ab·cos(C).
Pythagorean Triples
Integer solutions to a² + b² = c² are called Pythagorean triples. The most famous: 3-4-5 (9 + 16 = 25), 5-12-13 (25 + 144 = 169), 8-15-17, 7-24-25. Any multiple of a triple is also a triple: 6-8-10, 9-12-15. In construction, the 3-4-5 method is used to check right angles — measure 3 units on one wall, 4 on another, and if the diagonal is exactly 5, the corner is square.
Real-World Applications
Construction: calculating rafter length for a roof (rise and run give the rafter length). Navigation: shortest distance between two points on a grid. Screen sizes: a 65-inch TV means the diagonal is 65 inches. GPS distance calculation uses a 3D version (x² + y² + z²). Cable lengths: the hypotenuse gives the cable needed to reach from a pole to a ground anchor point. Stair handrail length: rise and horizontal run give the rail length.
Extension: Distance in 3D
In three dimensions, the distance formula extends Pythagoras: d = √(x² + y² + z²). The 3D diagonal of a box with dimensions l × w × h is √(l² + w² + h²). For example, the diagonal of a 3×4×5 box is √(9+16+25) = √50 = 7.07. This principle underlies coordinate geometry in any number of dimensions and is the basis of Euclidean distance used throughout mathematics, physics, and machine learning.
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