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)
Key Notes
- Formula: a² + b² = c²: In a right triangle, the sum of squares of the two legs equals the square of the hypotenuse. The hypotenuse is always opposite the right angle — always the longest side. The theorem fails for non-right triangles (use the Law of Cosines instead).
- Converse is equally important: If a² + b² = c² for the sides of a triangle, then the triangle is right-angled opposite c. This is used in construction to verify right angles: a 3-4-5 rope triangle creates a perfect 90° corner.
- Pythagorean triples: Integer solutions: (3,4,5); (5,12,13); (8,15,17); (7,24,25); (9,40,41). Any multiple of a triple also works: (6,8,10); (9,12,15). Infinitely many triples exist; they are parameterized by m²−n², 2mn, m²+n².
- Distance formula in coordinates: d = √((x₂−x₁)² + (y₂−y₁)²) is the Pythagorean theorem applied to a right triangle formed by the horizontal and vertical separations. In 3D: d = √((Δx)² + (Δy)² + (Δz)²). Each added dimension contributes another squared term.
- Applications: The Pythagorean theorem is used in construction (squaring foundations and frames), navigation (direct distance between two points), physics (vector magnitude), engineering (diagonal bracing), GPS (3D distance calculation), and is the foundation of Euclidean geometry and trigonometry.