Hypotenuse Calculator

The hypotenuse is the longest side of a right triangle, positioned directly opposite the right angle (90° angle). Core Definition: In any right triangle, the hypotenuse is: • The side opposite to the 90° angle • Always longer than either of the other two sides (legs) • Never forms the right angle itself Pythagorean Theorem: The fundamental formula relating the hypotenuse to the legs: c² = a² + b² or c = √(a² + b²) Where: • c = hypotenuse length • a, b = lengths of the two legs • This relationship ONLY applies to right triangles Calculation Steps: 1. Identify the two legs (sides forming the 90° angle) 2. Square each leg length: a² and b² 3. Add the squares: a² + b² 4. Take the square root: c = √(a² + b²) Example Calculation: If legs are 3 and 4: • c = √(3² + 4²) • c = √(9 + 16) • c = √25 • c = 5 Common Pythagorean Triples: Sets of three integers (a, b, c) that satisfy a² + b² = c²: • (3, 4, 5) — most common • (5, 12, 13) • (8, 15, 17) • (7, 24, 25) • Multiples also work: (6, 8, 10), (9, 12, 15), etc. Key Properties: 1. Length: c > a and c > b (always longest) 2. Angle opposite: Always 90° 3. Uniqueness: Each right triangle has exactly one hypotenuse 4. Measurement: Can be in any unit (feet, meters, inches, etc.) Etymology: From Greek "hypoteinousa" meaning "stretching under" (under the right angle) Applications: Construction (roof rafters, staircases), navigation (distance calculations), surveying, ladder safety (4:1 ratio), screen size measurements, and any problem involving right triangles.