Pythagorean Theorem
The Pythagorean theorem a² + b² = c² finds the length of any side of a right triangle.
The most famous formula in geometry.
The Formula
a² + b² = c²
In a right triangle, the square of the hypotenuse (the longest side) equals the sum of the squares of the other two sides.
This only works for right triangles (triangles with a 90° angle).
Variables
| Symbol | Meaning |
|---|---|
| a | One of the two shorter sides (a leg) |
| b | The other shorter side (a leg) |
| c | The hypotenuse (the side opposite the right angle — always the longest side) |
Solving for Each Side
- Find c: c = √(a² + b²)
- Find a: a = √(c² - b²)
- Find b: b = √(c² - a²)
Example 1
Find the hypotenuse of a right triangle with legs 3 and 4
a = 3, b = 4
c² = a² + b² = 3² + 4² = 9 + 16 = 25
c = √25
c = 5
Example 2
A ladder leans against a wall. The ladder is 13 m long and its base is 5 m from the wall. How high does it reach?
c = 13 (ladder = hypotenuse), a = 5 (base), b = ? (height)
b² = c² - a² = 13² - 5² = 169 - 25 = 144
b = √144
b = 12 m
Common Pythagorean Triples
- 3, 4, 5 (and multiples: 6-8-10, 9-12-15, etc.)
- 5, 12, 13
- 8, 15, 17
- 7, 24, 25
When to Use It
Use the Pythagorean theorem when:
- Finding the missing side of a right triangle
- Checking if a triangle is a right triangle (test if a² + b² = c²)
- Calculating distances in 2D space (the distance formula is based on this)
- Solving real-world problems involving right angles (ladders, ramps, diagonals)