Hypotenuse Calculator
Solve the hypotenuse of a right triangle from the two legs (a² + b² = c²), or solve a missing leg given the hypotenuse and one leg.
Last updated: September 2026
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About this calculator
The Pythagorean theorem, from Euclid Book I Proposition 47 (attributed to Pythagoras of Samos, c. 570–495 BCE), states that for any right triangle with legs a and b and hypotenuse c: a² + b² = c². The hypotenuse is the side opposite the right angle and always the longest side.
To solve for the hypotenuse: c = √(a² + b²). To solve for a missing leg given the hypotenuse and one leg: b = √(c² − a²). If c < a, the geometry is impossible — the hypotenuse must be the longest side of a right triangle.
Common Pythagorean triples (integer-leg right triangles): (3, 4, 5), (5, 12, 13), (8, 15, 17), (7, 24, 25), (20, 21, 29). Carpenters use the 3-4-5 rule to check that a corner is square: measure 3 ft along one wall, 4 ft along the perpendicular wall, and if the diagonal is exactly 5 ft, the corner is 90°. Multiples (6-8-10, 9-12-15) work the same way over larger distances.
The theorem generalises to distance in the plane (d = √((x2−x1)² + (y2−y1)²)), 3D space (d = √(Δx² + Δy² + Δz²)), and to inner products in any Euclidean space. Non-Euclidean geometries (spherical, hyperbolic) have different distance formulas but reduce to Pythagoras in the small-triangle limit.
How to use
Example — hypotenuse of a 3-4 right triangle. c = √(3² + 4²) = √(9 + 16) = √25 = 5. The classic 3-4-5 triple. Example — missing leg given c = 13, a = 5. b = √(13² − 5²) = √(169 − 25) = √144 = 12. The 5-12-13 triple. If you enter a > c, the answer is 0 because the geometry is impossible (the leg cannot exceed the hypotenuse of a right triangle).
Frequently asked questions
Does the theorem work for non-right triangles?
No — the Pythagorean theorem applies only to right triangles. For a general triangle, use the Law of Cosines: c² = a² + b² − 2ab·cos(C). When C = 90°, cos(C) = 0 and the formula reduces to a² + b² = c².
What if I have the two acute angles instead of two legs?
You need at least one side length. Two angles alone define the triangle's shape but not its size — a 30-60-90 triangle can have any hypotenuse. Given one side and one non-right angle, use trigonometry: side = hypotenuse × sin(angle) or hypotenuse × cos(angle) depending on which side you have.
Are the answers always rational?
No — the hypotenuse of a 1-1 right triangle is √2, an irrational number. The historical discovery of √2 is credited to the Pythagorean school and reportedly caused a crisis about the nature of numbers. Only special integer triples (3-4-5 etc.) give rational hypotenuses from integer legs.
What are the units?
Whatever you input — feet in, feet out; meters in, meters out. The calculator is unit-agnostic. Make sure the two legs use the same units before entering, or you'll get a nonsense answer.