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 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.