Pythagorean Theorem Calculator
Hypotenuse or missing leg for right triangles, with check.
What it does
Right triangles obey one perfect rule: the legs squared add to the hypotenuse squared. That single equation sizes ladders, diagonals, screen sizes, and distances that hide right angles.
The tool above solves either direction: hypotenuse from two legs, or a missing leg from hypotenuse and one leg, each with the check math shown.
Integer triples like 3-4-5 and 5-12-13 fall out exactly, which carpenters exploit with 3-4-5 strings to square foundations without a single angle tool.
The converse also works: sides satisfying the equation guarantee a right angle. Inspectors use it to verify square corners from tape measures alone.
How to use
- Pick which side is missing.
- Enter the two known sides with matching units.
- Read the answer with its check line.
- Confirm the right angle exists in the real shape.
Formula
- a, b
- Legs meeting at the right angle.
- c
- Hypotenuse opposite the right angle, longest side.
Units carry through untouched: feet in gives feet out. Areas scale differently and need their own math.
Worked example
Example: legs 3 and 4
Right triangle with legs 3 and 4.
- Squares: 3² + 4² = 9 + 16 = 25.
- Root: sqrt(25) = 5.
- Answer: hypotenuse 5.
- Reverse check: leg from 5 and 3 is sqrt(25-9) = 4.
Reference table
| Legs | Hypotenuse | Use |
|---|---|---|
| 3 - 4 | 5 | Small square checks |
| 5 - 12 | 13 | Room diagonals |
| 6 - 8 | 10 | Scaled 3-4-5 |
| 9 - 12 | 15 | Scaled 3-4-5 |
| 20 - 21 | 29 | Long runs |
Limits
- Right triangles only. Others need law of cosines.
- Flat geometry assumed.
- Measurement error passes straight through.
- Huge numbers lose floating digits.
Common mistakes
- Adding legs instead of squares.
- Forgetting the final square root.
- Calling the longest leg the hypotenuse wrongly.
- Mixing units between the two legs.
- Applying to non right triangles.
Try it yourself
Quick checks. Answers sit tucked under each question.
Legs 6 and 8?
36 + 64 = 100. Root 10.
Hypotenuse 13, leg 5?
169 - 25 = 144. Root 12.
Questions
What is the Pythagorean theorem?
Legs squared add to hypotenuse squared in right triangles.
How do builders use 3-4-5?
Measure 3 and 4 along walls; a 5 diagonal proves square corners.
What are Pythagorean triples?
Whole number sets like 3-4-5 and 5-12-13 satisfying the equation.
Can I find angles too?
Yes with inverse trig on the sides, outside this page scope.
Does it work in 3D?
Extended: space diagonal squares sum all three edges squared.
Why is hypotenuse longest?
It spans the full right angle while legs span parts.