Sprint 8 Converter + Math
Pythagorean Theorem Solver
Right triangle hypotenuse.
Hypotenuse
10.1980
How it works
The Pythagorean theorem states that in a right triangle, the square of the hypotenuse (the side opposite the right angle) equals the sum of the squares of the other two sides: a² + b² = c². This is one of the most useful relationships in geometry, with applications spanning construction, navigation, physics, and computer graphics.
**What the solver computes** Enter any two sides — the solver computes the third. Inputs: legs a and b (adjacent to the right angle), or one leg and hypotenuse c. Output: the missing side, rounded to the precision you specify, with exact symbolic form if the result is a perfect square.
**Pythagorean triples** Integer solutions to a² + b² = c² are called Pythagorean triples. The most famous: (3, 4, 5), (5, 12, 13), (8, 15, 17), (7, 24, 25). Multiples also work: (6, 8, 10), (9, 12, 15). Builders use the (3, 4, 5) triple to create perfect right angles: measure 3 units along one wall, 4 along the other — if the diagonal is 5, the corner is square.
**Extensions and applications** The 3D distance formula extends the theorem: distance = √(Δx² + Δy² + Δz²). In coordinate geometry, the distance between two points (x₁, y₁) and (x₂, y₂) is √((x₂−x₁)² + (y₂−y₁)²) — the 2D version. Computer graphics use this constantly for collision detection, ray-casting, and shader calculations.
**Proof overview** The visual proof: four copies of the right triangle arranged inside a square of side c create a square of side (a−b) in the centre, proving c² = a² + b² by area. Over 370 distinct proofs have been published.
Privacy: all computation runs in the browser. No data is transmitted.
Frequently Asked Questions
- A Pythagorean triple is a set of three positive integers (a, b, c) satisfying a² + b² = c². Common ones: (3,4,5): 9+16=25 ✓; (5,12,13): 25+144=169 ✓; (8,15,17): 64+225=289 ✓; (7,24,25); (20,21,29). Multiples also work: (6,8,10), (9,12,15), (12,16,20). Builders use the 3-4-5 triple to verify right angles — measure 3 units from a corner along one wall and 4 units along the other; if the diagonal is 5, the corner is perfectly square.
- GPS receivers calculate their position by trilateration: measuring their distance from multiple satellites. Distance from one satellite = c × t, where t is signal travel time. In 2D, distance to a point equals √((x₂−x₁)² + (y₂−y₁)²) — the Pythagorean theorem. In 3D navigation, the 3D distance formula (√(Δx² + Δy² + Δz²)) extends this to three dimensions. Every turn-by-turn navigation calculation involves thousands of distance computations per second, all rooted in the Pythagorean theorem.
- No — the Pythagorean theorem is specific to flat (Euclidean) geometry. On the surface of a sphere (like Earth), the sum of squares of sides of a 'right triangle' is not equal to the square of the hypotenuse — spherical excess changes the relationship. The spherical law of cosines replaces it. In relativity, the spacetime interval uses a 'Minkowski metric' where time and space have opposite signs: s² = −(ct)² + x² + y² + z² — a hyperbolic geometry. GPS systems must account for both spherical Earth geometry and relativistic corrections.
- In a right triangle, the right angle (90°) is the largest angle. The side opposite the largest angle is always the longest side (a fundamental property of triangles: larger angles face longer sides). Since c is opposite the 90° angle, c > a and c > b always. Mathematically: c² = a² + b² > a² → c > a, and similarly c > b. The hypotenuse is therefore always longer than either leg, and their difference shrinks as the triangle approaches isosceles (45-45-90).