How to find the hypotenuse
The hypotenuse is the longest side of a right triangle, opposite the 90° angle. The Pythagorean theorem links it to the two other sides (the legs):
a² + b² = c² so c = √(a² + b²)
- Square each leg.
- Add the two squares.
- Take the square root of the sum.
Example: legs of 6 and 8 → 36 + 64 = 100 → c = √100 = 10.
Common Pythagorean triples
| a | b | c |
|---|---|---|
| 3 | 4 | 5 |
| 5 | 12 | 13 |
| 8 | 15 | 17 |
| 7 | 24 | 25 |
| 20 | 21 | 29 |
Real-world uses
- Construction: the 3-4-5 rule checks that a corner is square — measure 3 ft and 4 ft along two walls; the diagonal should be exactly 5 ft.
- Ladders and ramps: the ladder is the hypotenuse; the wall height and the distance from the wall are the legs.
- Screens: a TV's diagonal size is the hypotenuse of its width and height.
More maths tools: the quadratic formula calculator and the scientific calculator.
Special right triangles
| Triangle | Side ratio | Example |
|---|---|---|
| 45° – 45° – 90° | 1 : 1 : √2 | Legs 5 and 5 → hypotenuse 7.071 |
| 30° – 60° – 90° | 1 : √3 : 2 | Short leg 4 → long leg 6.928, hypotenuse 8 |
Finding the hypotenuse from one side and an angle
If you know one leg and an acute angle instead of both legs, use trigonometry: c = a ÷ sin(α), where α is the angle opposite leg a, or c = b ÷ cos(α). With a = 5 and α = 30°: c = 5 ÷ 0.5 = 10. The scientific calculator gives sin and cos values.
Worked example: a roof rafter
A roof rises 2.4 m over a horizontal run of 4.5 m. The rafter length is the hypotenuse: √(2.4² + 4.5²) = √(5.76 + 20.25) = √26.01 = 5.1 m, before adding the overhang. The roof angle is atan(2.4 ÷ 4.5) = 28.1°. Builders use the same calculation for stair stringers and wheelchair ramps.
The distance formula between two points is the same theorem applied to coordinates — see the distance formula calculator. To read values from engineering tables, try the interpolation calculator.