Calculation details
Planning estimate only, not financial, tax, or legal advice. Verify assumptions and current rules before making decisions.
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What does this calculator estimate?
The Pythagorean theorem calculator finds the missing side of a right triangle. Choose whether you know the two legs or the hypotenuse and one leg, enter the values, and get the third side.
- a² + b² = c² (c = hypotenuse)
- Hypotenuse: c = √(a² + b²)
- Leg: a = √(c² − b²)
What the theorem says
In a right triangle, the square of the hypotenuse equals the sum of the squares of the legs. It's the foundation of distance measurement in 2D: the straight-line distance between two points is the hypotenuse of the right triangle they define.
Limitations to watch for
The theorem only applies to right triangles. When solving for a leg, the known side (hypotenuse) must be longer than the given leg or the result is undefined. Units must match — mixing feet and inches invalidates the result.
How to use it in practice
Use it for distances, ladder problems, screen diagonals, roof slopes, and navigation offsets. When solving for a leg, make sure you enter the hypotenuse as the larger value. For 3D distances, apply the theorem twice.
["Choose what you're solving for: hypotenuse or a leg.", 'Enter the two known side lengths.', 'The tool applies a² + b² = c² to find the missing side.']
How this calculator works
Formula
a² + b² = c², where c is the hypotenuse. Solving for the hypotenuse: c = √(a² + b²). Solving for a leg: a = √(c² − b²) or b = √(c² − a²).
Worked example
A right triangle with legs 3 and 4 has hypotenuse c = √(9 + 16) = √25 = 5. Given hypotenuse 13 and leg 5, the other leg is √(169 − 25) = √144 = 12.
Assumptions to verify
- The triangle is right-angled.
- All side lengths are in the same units.
- The known leg is shorter than the hypotenuse when solving for a leg.
Frequently asked questions
What is the Pythagorean theorem?
a² + b² = c²: in a right triangle, the square of the hypotenuse equals the sum of the squares of the other two sides.
How do I find the hypotenuse?
Take the square root of the sum of the squares: c = √(a² + b²).
How do I find a missing leg?
Subtract the square of the known leg from the square of the hypotenuse, then take the root: a = √(c² − b²).
Does it work for non-right triangles?
No — the theorem applies only to right triangles. Other triangles use the law of cosines.
Why is the hypotenuse the longest side?
Because it's opposite the right angle (90°), and the side opposite the largest angle is always the longest in any triangle.
What are common applications?
Computing straight-line distances, ladder reach, TV/screen diagonals, roof pitches, and navigation offsets.
What if the result is undefined?
If the known leg is longer than the hypotenuse you entered, no real triangle exists — check that the hypotenuse is the largest value.
Cite this tool
BoringToolsKit. “Pythagorean Theorem Calculator.” boringtoolskit.com/pythagorean-theorem-calculator/ (reviewed 2026-08-25). Free to reference in articles, syllabi, and answer posts with a link.
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