This Pythagorean theorem calculator solves one missing side of a right triangle. Enter legs a and b and hypotenuse c, with exactly one value set to 0 as the unknown. The default uses a=3, b=4, and c=0, which solves to hypotenuse 5.
When you always have both legs and only need the hypotenuse, the right triangle calculator is a shorter path. For area from three sides, use the triangle calculator.
In a right triangle, a² + b² = c², where c is the hypotenuse (the side opposite the right angle). To find the hypotenuse, use c = √(a² + b²): legs 3 and 4 give √25 = 5. To find a missing leg, use a = √(c² − b²): hypotenuse 5 and leg 4 give 3. In this calculator, enter 0 in the side you want to find.
How the formula works
If c is zero, return √(a² + b²). If a is zero, return √(c² – b²). If b is zero, return √(c² – a²). The hypotenuse must be longer than each known leg when a leg is missing.
Worked example
a = 3, b = 4, c = 0. Missing hypotenuse = √(9 + 16) = √25 = 5.
| Input | Value |
|---|---|
| Leg a | 3 |
| Leg b | 4 |
| Hypotenuse c | 0 (solve) |
| Missing side | 5 |
Step-by-step examples
| Problem | Steps | Answer |
|---|---|---|
| Legs 3 and 4 | 9 + 16 = 25; √25 | c = 5 |
| Hypotenuse 5, leg 4 | 25 − 16 = 9; √9 | a = 3 |
| Legs 1 and 1 | 1 + 1 = 2; √2 | c ≈ 1.414214 |
| Room 10 × 12 ft, corner to corner | 100 + 144 = 244; √244 | ≈ 15.62 ft |
| 12 ft ladder, base 3 ft from wall | 144 − 9 = 135; √135 | reaches ≈ 11.62 ft high |
| 55 in 16:9 TV | width = 55 × 16/√(16² + 9²); height = 55 × 9/√337 | ≈ 47.94 × 26.96 in |
Common Pythagorean triples
| a | b | c |
|---|---|---|
| 3 | 4 | 5 |
| 6 | 8 | 10 |
| 5 | 12 | 13 |
| 9 | 12 | 15 |
| 8 | 15 | 17 |
| 7 | 24 | 25 |
| 20 | 21 | 29 |
How to use the fields
- Leg a: enter the length, or 0 to solve for a.
- Leg b: enter the length, or 0 to solve for b.
- Hypotenuse c: enter the length, or 0 to solve for c.
- Mark the zero field before you type, and leave only one side at zero.
Solving a missing leg
Example pattern: a=0, b=4, c=5 yields a=3. Always confirm c > b before trusting the square root of c² – b².
Common mistakes
- Leaving two zeros instead of one
- Putting the longest side in a leg field when it is the hypotenuse
- Using the theorem on non-right triangles
- Mixing units across sides
Real-world uses
Carpenters still use 3-4-5 checks to square a corner. Measure 3 and 4 along the edges and confirm the diagonal is 5 in matching units. On a map or grid, horizontal and vertical runs form the legs and the straight line distance is the hypotenuse: enter the runs as a and b with c=0, or put a zero in a leg field if you know the straight distance and one run. For distance between two points, the distance calculator does the same math from coordinates.
Non integer sides
Real cuts are rarely perfect integers, so enter decimals in matching units and expect a decimal hypotenuse. Keep full precision through the square root and round once, as in the legs 1 and 1 example (√2 ≈ 1.414214).
Limitations
Only right triangles with exactly one unknown side are supported. Angles, altitude to the hypotenuse, and oblique triangles are out of scope.