Pythagorean Theorem Calculator
Calculate any side of a right triangle using the Pythagorean theorem (a² + b² = c²). Find hypotenuse or missing leg instantly.
1x² − 5x + 6 = 0
Discriminant = 1
two real roots
x = (−b ± √(b²−4ac)) / 2a
x₁ = 3
x₂ = 2
About the Pythagorean Theorem Calculator
The Pythagorean theorem — a² + b² = c² — is arguably the most famous equation in mathematics, relating the three sides of any right triangle. Known in ancient Babylon and Egypt and formally proved by the Greeks, it underpins everything from construction and navigation to signal processing and spacetime geometry. Our Pythagorean theorem calculator solves for any missing side given the other two.
Formula
a² + b² = c² (c = hypotenuse, a and b = legs)
How It Works
Given legs a and b: hypotenuse c = √(a² + b²). Given hypotenuse c and leg a: other leg b = √(c² − a²). The calculator also verifies whether three given sides form a valid right triangle and identifies Pythagorean triples (integer solutions like 3-4-5, 5-12-13, 8-15-17). It handles any units — the theorem is unit-agnostic.
Tips & Best Practices
- ✓The 3-4-5 right triangle is used by builders to create perfect 90° corners on foundations.
- ✓Pythagorean triples: 3-4-5, 5-12-13, 7-24-25, 8-15-17, 9-40-41, 11-60-61.
- ✓In 3D space, the distance formula extends to: d = √(Δx² + Δy² + Δz²).
- ✓The theorem fails for non-Euclidean geometry (spherical or hyperbolic surfaces).
- ✓GPS systems use multi-dimensional extensions of the Pythagorean theorem for positioning.
Who Uses This Calculator
Carpenters squaring foundations, students learning trigonometry, engineers calculating cable lengths, game developers implementing collision detection, and anyone who needs to find the straight-line distance between two points in a coordinate plane use the Pythagorean theorem constantly.
Optimised for: USA · Canada · UK · Australia · Calculations run in your browser · No data stored
Frequently Asked Questions
What is the Pythagorean theorem?
For a right triangle: a² + b² = c², where c is the hypotenuse. If a=3 and b=4, then c=5 (the famous 3-4-5 triangle).