Pythagorean Theorem
The Pythagorean theorem relates the sides of a right triangle: a² + b² = c².
Learn the formula with worked examples.
The Formula
In any right triangle, the square of the hypotenuse (the side opposite the right angle) equals the sum of the squares of the other two sides.
This theorem was known to the ancient Babylonians around 1800 BCE, but it is named after the Greek mathematician Pythagoras (circa 570–495 BCE).
Variables
| Symbol | Meaning |
|---|---|
| a | One leg of the right triangle |
| b | The other leg of the right triangle |
| c | The hypotenuse (longest side, opposite the right angle) |
Useful Rearrangements
a = √(c² − b²)
b = √(c² − a²)
Example 1 — Finding the Hypotenuse
A right triangle has legs of 3 cm and 4 cm. Find the hypotenuse.
Known values: a = 3 cm, b = 4 cm
c² = a² + b² = 3² + 4² = 9 + 16 = 25
c = √25
c = 5 cm
Example 2 — Finding a Missing Leg
A ladder 13 feet long leans against a wall. The base is 5 feet from the wall. How high up does the ladder reach?
Known values: c = 13 ft (ladder), b = 5 ft (base distance)
a² = c² − b² = 13² − 5² = 169 − 25 = 144
a = √144
a = 12 ft
Example 3 — Distance Between Two Points
Find the straight-line distance between points (1, 2) and (4, 6) on a coordinate plane.
Horizontal distance: a = 4 − 1 = 3
Vertical distance: b = 6 − 2 = 4
d = √(3² + 4²) = √(9 + 16) = √25
d = 5 units
Common Pythagorean Triples
| a | b | c |
|---|---|---|
| 3 | 4 | 5 |
| 5 | 12 | 13 |
| 8 | 15 | 17 |
| 7 | 24 | 25 |
Any multiple of a Pythagorean triple is also a valid triple. For example, 6-8-10 is simply 2× the 3-4-5 triple.
When to Use It
- Finding the diagonal of a rectangle or screen
- Calculating straight-line distances on a map or coordinate plane
- Construction — checking if a corner is exactly 90° (the 3-4-5 rule)
- Any problem involving right triangles