What Is the Pythagorean Theorem?
The Pythagorean theorem is one of the most fundamental relationships in geometry. For any right triangle with legs a and b and hypotenuse c:
c² = a² + b²
The hypotenuse is the longest side — always opposite the 90° angle. If you know any two sides, you can find the third.
How to Find the Hypotenuse
Square both legs, add them, and take the square root:
c = √(a² + b²)
Example: Legs of 6 cm and 8 cm. c = √(36 + 64) = √100 = 10 cm
How to Find a Missing Leg
Rearrange the formula by subtracting the known square from the hypotenuse squared:
a = √(c² − b²)
Example: Hypotenuse 15 cm, one leg 9 cm. a = √(225 − 81) = √144 = 12 cm
Angles, Area, and Perimeter
Once you have all three sides the other measurements follow directly:
- Angle α (opposite leg a): α = arctan(a ÷ b)
- Angle β (opposite leg b): β = 90° − α
- Area: A = (a × b) ÷ 2
- Perimeter: P = a + b + c
Example: a = 3, b = 4, c = 5. Area = (3 × 4) ÷ 2 = 6 cm² · Perimeter = 3 + 4 + 5 = 12 cm
Common Right Triangle Examples
| Leg a | Leg b | Hypotenuse c |
|---|---|---|
| 3 | 4 | 5 |
| 5 | 12 | 13 |
| 8 | 15 | 17 |
| 7 | 24 | 25 |
| 9 | 40 | 41 |
These are called Pythagorean triples — sets of whole numbers that satisfy c² = a² + b² exactly. The 3-4-5 triple is the most commonly used in construction and carpentry for checking right angles.