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Pythagorean Theorem Calculator

Pythagorean Theorem Calculator

Solve right triangle problems using the Pythagorean theorem. Find missing sides with step-by-step solutions and learn the mathematical concepts behind this fundamental theorem.

100% FreeStep-by-step SolutionsVisual Triangle
Pythagorean Theorem Calculator
Enter any two sides of a right triangle to find the third side: a2+b2=c2a^2 + b^2 = c^2
a2+b2=c2a^2 + b^2 = c^2

where a and b are legs, and c is the hypotenuse

First leg of the triangle

Second leg of the triangle

Hypotenuse (longest side)

Instructions: Enter any two sides to calculate the third. Leave one field empty for the side you want to find.
Try These Examples
Click on any example to automatically fill the calculator
Example
a = 3
b = 4
c = ?

Classic 3-4-5 triangle

Example
a = 5
b = ?
c = 13

5-12-13 Pythagorean triple

Example
a = ?
b = 8
c = 10

Find missing leg

Example
a = 1
b = 1
c = ?

45-45-90 triangle

What is the Pythagorean Theorem?

The Pythagorean theorem states that in a right triangle, the square of the hypotenuse (the side opposite the right angle) equals the sum of squares of the other two sides.

a2+b2=c2a^2 + b^2 = c^2

Key Components:

  • a, b: The legs (sides forming the right angle)
  • c: The hypotenuse (longest side, opposite right angle)
  • Right Triangle: Must have a 90° angle

Historical Note: While named after Pythagoras (c. 570-495 BC), this theorem was known to ancient civilizations including Babylonians and Indians centuries earlier.

How to Apply the Theorem

Three Cases:

Finding the hypotenuse (c):
c=a2+b2c = \sqrt{a^2 + b^2}
Finding a leg (a):
a=c2−b2a = \sqrt{c^2 - b^2}
Finding a leg (b):
b=c2−a2b = \sqrt{c^2 - a^2}

Common Pythagorean Triples:

3-4-5, 5-12-13, 8-15-17, 7-24-25
9-12-15, 12-16-20, 15-20-25
And their multiples...
Real-World Applications

Construction & Architecture

  • Calculating diagonal braces in framing
  • Determining rafter lengths for roofs
  • Ensuring square corners in foundations
  • Planning staircase dimensions

Engineering & Design

  • Navigation and GPS calculations
  • Computer graphics and game development
  • Robotics and movement planning
  • Structural engineering analysis

Everyday Life

  • Calculating TV screen sizes
  • Finding shortest distances on maps
  • Sports field measurements
  • Home improvement projects
Famous Proofs of the Pythagorean Theorem

Euclid's Proof (Geometric)

Uses the areas of squares constructed on each side of the triangle, showing that the area of the square on the hypotenuse equals the sum of areas of squares on the other two sides.

President Garfield's Proof

Uses a trapezoid formed by arranging two copies of the right triangle, comparing its area calculated in two different ways to derive the theorem.

Chinese Proof (Zhao Shuang)

Arranges four congruent right triangles around a central square, showing the relationship through area calculations.

Algebraic Proof

Uses coordinate geometry and the distance formula to prove the theorem algebraically, connecting it to analytic geometry.

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