Pythagorean Theorem Calculator
Solve a² + b² = c² for any missing side — find the hypotenuse from two legs, or a leg from the hypotenuse and one leg.
What is a pythagorean theorem calculator?
The Pythagorean theorem states that in any right triangle, the square of the hypotenuse equals the sum of the squares of the two legs: a² + b² = c².
This calculator solves for whichever side you don't enter. It also returns the perimeter, area, and the two acute angles.
Formula
Solve for the missing side:
- Hypotenuse: c = √(a² + b²)
- Leg a: a = √(c² − b²)
- Leg b: b = √(c² − a²)
The angles are found by:
- Angle A = arctan(a / b)
- Angle B = arctan(b / a)
- Angle A + Angle B = 90°
Worked example
A 3-4-5 right triangle: 3² + 4² = 9 + 16 = 25 = 5².
For a = 6, b = 8: c = √(36 + 64) = √100 = 10. Same shape, scaled up.
How to use this calculator
- Enter the two known sides.
- Leave the unknown side blank — the calculator detects which one is missing and solves for it.
- Perimeter, area, and acute angles update automatically.
Frequently asked questions
When does the theorem apply?
Only to right triangles — triangles with one 90° angle. For any other triangle, use the law of cosines instead.
What's a Pythagorean triple?
Three positive integers (a, b, c) that satisfy a² + b² = c². Famous ones: (3, 4, 5), (5, 12, 13), (8, 15, 17), (7, 24, 25).
Where is this used in real life?
Construction (squaring a foundation), navigation (straight-line distance), packing (will it fit diagonally?), and screen sizes (a 65" TV is the diagonal, not the width).
Does it work in 3D?
Yes — the 3D distance between two points is √(Δx² + Δy² + Δz²), an extended Pythagorean theorem. Use the distance formula calculator for 2D point-to-point distance.