Hypotenuse Calculator
Calculate the hypotenuse of a right triangle from its two legs instantly. Enter any two side lengths to get the exact result, free online.
How to Use the Hypotenuse Calculator
- Enter the length of Side a.
- Enter the length of Side b.
- View the hypotenuse instantly.
Features of the Hypotenuse Calculator
- Instantly calculates the hypotenuse of a right triangle from its two legs.
- Works with whole numbers and decimals.
- Live results as you type, with no separate calculate step required.
- Free to use with no sign-up required.
Hypotenuse Formula
The Pythagorean theorem states that in a right triangle, the square of the hypotenuse equals the sum of the squares of the other two sides.
c = √(a² + b²)
Where a and b are the two legs of the right triangle, and c is the hypotenuse, the side opposite the right angle.
Hypotenuse Reference Table
| Side a | Side b | Hypotenuse (c) |
|---|---|---|
| 3 | 4 | 5 |
| 5 | 12 | 13 |
| 6 | 8 | 10 |
| 8 | 15 | 17 |
| 7 | 24 | 25 |
| 9 | 12 | 15 |
How to Calculate the Hypotenuse
To find the hypotenuse, square both leg lengths, add the results together, then take the square root of the sum.
Suppose Side a is 5 and Side b is 12. c = √(5² + 12²) = √(25 + 144) = √169 = 13.
Thus, the hypotenuse is 13.
Solved Examples on Finding the Hypotenuse
1. The Classic 3-4-5 Triangle
Example: Side a = 3, Side b = 4.
c = √(3² + 4²) = √(9 + 16) = √25 = 5.
2. A Larger Right Triangle
Example: Side a = 8, Side b = 15.
c = √(8² + 15²) = √(64 + 225) = √289 = 17.
3. A Right Triangle With a Non-Perfect-Square Result
Example: Side a = 5, Side b = 7.
c = √(5² + 7²) = √(25 + 49) = √74 ≈ 8.6023.
4. Equal-Length Legs
Example: Side a = 6, Side b = 6.
c = √(6² + 6²) = √72 ≈ 8.4853.
1. What is the hypotenuse of a right triangle?
The hypotenuse is the longest side of a right triangle, located opposite the right angle. It is always longer than either of the other two sides, called legs.
2. Does the Pythagorean theorem work for all triangles?
No, it only applies to right triangles, meaning triangles with one 90-degree angle. For other triangle types, different formulas like the Law of Cosines are used instead.
3. What are Pythagorean triples?
Pythagorean triples are sets of three positive whole numbers that satisfy a² + b² = c², such as (3, 4, 5), (5, 12, 13), and (8, 15, 17). These combinations always produce a whole-number hypotenuse.
4. Can the legs be decimal values?
Yes, the legs can be any positive number, including decimals. The resulting hypotenuse simply may not be a whole number in that case.
5. Can a leg be zero or negative?
No. Both legs must be positive numbers, since a triangle can’t have a side of zero or negative length.
6. Where is the hypotenuse formula used in practice?
The hypotenuse formula is used in construction for calculating diagonal supports and roof rafters, in navigation for finding the shortest straight-line distance, and in physics for combining perpendicular vector components.