Pythagorean Theorem Calculator

Calculate the hypotenuse or a missing leg using a² + b² = c² with full step-by-step substitution.

Choose What to Find

Result

Hypotenuse c
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units

Step-by-Step Solution

a² + b² = c²

The Pythagorean Theorem

The Pythagorean theorem is one of the most fundamental theorems in mathematics. It states that in a right-angled triangle, the square of the length of the hypotenuse (the side opposite the right angle) is equal to the sum of the squares of the lengths of the other two sides: a² + b² = c².

This theorem is attributed to the ancient Greek mathematician Pythagoras, though it was known to Babylonian mathematicians over a thousand years earlier.

How to Use the Pythagorean Theorem

Find the Hypotenuse

Given two legs a and b, the hypotenuse is:

c = sqrt(a² + b²)

Find a Missing Leg

Given the hypotenuse c and one leg a:

b = sqrt(c² - a²)

Triangle Properties

Area and perimeter from the three sides:

Area = (a x b) / 2

Proofs of the Pythagorean Theorem

There are hundreds of known proofs of the Pythagorean theorem, including geometric proofs, algebraic proofs, and proofs using similar triangles. The most famous geometric proof involves rearranging four right triangles within a square.

Applications

  • Navigation: Calculating straight-line distances between two points.
  • Construction: Ensuring walls are square using the 3-4-5 method.
  • Computer Graphics: Calculating distances between pixels or points in 2D/3D space.
  • Physics: Resolving vector components and calculating resultant forces.

Extensions

  • The distance formula in 2D extends Pythagoras: d = sqrt((x2-x1)² + (y2-y1)²).
  • In 3D space: d = sqrt(dx² + dy² + dz²).
  • The law of cosines generalizes the theorem for any triangle: c² = a² + b² - 2ab*cos(C).