📐 Pythagorean Theorem Calculator

The Pythagorean theorem states that in a right triangle, the square of the length of the hypotenuse (c) equals the sum of the squares of the other two sides (a and b). Mathematically: a² + b² = c². This fundamental relationship only applies to triangles with a 90-degree angle.

No, the Pythagorean theorem only works for right triangles (triangles with one 90° angle). For other triangles, you need to use the Law of Cosines: c² = a² + b² - 2ab·cos(C), which is a generalization of the Pythagorean theorem.

A Pythagorean triple is a set of three positive integers (a, b, c) that satisfy the equation a² + b² = c². The most famous example is 3-4-5. Other common triples include 5-12-13, 8-15-17, and 7-24-25. Any multiple of a Pythagorean triple is also a valid triple.

To find the hypotenuse (c), use the formula c = √(a² + b²). Square both legs, add them together, then take the square root of the sum. For example, if a = 3 and b = 4: c = √(9 + 16) = √25 = 5.

To find a missing leg when you know the hypotenuse and one leg, rearrange the formula. For leg a: a = √(c² - b²). For leg b: b = √(c² - a²). Remember that the hypotenuse must always be longer than either leg.

The converse states: If a² + b² = c² for a triangle with sides a, b, and c (where c is the longest side), then the triangle is a right triangle. This is useful for verifying whether three given lengths can form a right triangle.

While named after the Greek mathematician Pythagoras (c. 570–495 BC), evidence shows the theorem was known earlier. Babylonian clay tablets from 1800 BC demonstrate knowledge of Pythagorean triples. The theorem was also independently discovered in ancient India and China.

Yes! The 3D distance formula extends the theorem: d = √(x² + y² + z²). This calculates the diagonal distance through a rectangular box or the distance between two points in 3D space, essential in physics, engineering, and computer graphics.