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Pythagorean Theorem Calculator

Calculate the missing side of a right triangle using the Pythagorean theorem. Legs and hypotenuse.

Sources: Khan Academy -- Pythagorean theorem · Wolfram MathWorld -- Pythagorean Theorem

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How to Use the Pythagorean Theorem

This calculator applies the Pythagorean theorem to find the missing side of a right triangle. Enter any two sides and get the third, along with the triangle's area and perimeter.

The Formula

The Pythagorean theorem states that in a right triangle:

a² + b² = c²

where a and b are the legs (the sides forming the right angle) and c is the hypotenuse (the longest side, opposite the right angle).

From this formula we derive:

  • Hypotenuse: c = sqrt(a² + b²)
  • Leg: a = sqrt(c² - b²)

Pythagorean Triples

Pythagorean triples are sets of three positive integers that satisfy the theorem. The most famous: (3, 4, 5), (5, 12, 13), (8, 15, 17), and (7, 24, 25). Every multiple of a Pythagorean triple is also a triple.

History of the Theorem

Although the theorem bears the name of Pythagoras of Samos (6th century BC), the relationship was known to the Babylonians at least from 1800 BC -- the clay tablet Plimpton 322 contains Pythagorean triples. Over 400 different proofs of the theorem are known today.

Everyday Applications

The Pythagorean theorem is constantly used in construction (verifying right angles with the 3-4-5 technique), navigation (calculating straight-line distances), computing (distance between two points on screen), surveying, and architecture.

Frequently Asked Questions

What is the Pythagorean theorem?▼
The Pythagorean theorem states that in a right triangle, the square of the hypotenuse equals the sum of the squares of the two legs: a² + b² = c². It is one of the most important theorems in mathematics, with over 400 known proofs.
What are Pythagorean triples?▼
Pythagorean triples are sets of three positive integers (a, b, c) that satisfy a² + b² = c². The most well-known are (3, 4, 5), (5, 12, 13), (8, 15, 17), (7, 24, 25). Every multiple of a Pythagorean triple is also a triple.
Does the theorem only apply to right triangles?▼
Yes, the Pythagorean theorem applies exclusively to right triangles. For non-right triangles, the law of cosines (a generalization of Pythagoras) is used: c² = a² + b² - 2ab*cos(C), where C is the angle between sides a and b.