About the Pythagorean Theorem Calculator
This Pythagorean theorem calculator finds the missing side of any right triangle. Choose whether you need the hypotenuse (c) or one of the legs (a or b), enter the two sides you know, and it solves a² + b² = c² for you, showing the working along with both acute angles, the area and the perimeter.
Students use it to check homework, and tradespeople use it constantly: checking that a wall corner is square with the 3-4-5 rule, finding the length of a ladder or rafter, the diagonal of a screen or room, or how far a ramp will run.
The theorem only applies to right triangles — one angle must be exactly 90°. When you solve for a leg, the hypotenuse must be longer than the other leg, otherwise no right triangle exists and the calculator tells you so.
With the default inputs, the missing side is 13. Change any value above to recalculate instantly.
How to use the pythagorean theorem calculator
- 1Choose which side is missing.
- 2Enter the two sides you know, in the same unit.
- 3Read the missing side and the working in the notes.
- 4Use the angles, area and perimeter for follow-up questions.
Formula and method
In a right triangle, the square on the hypotenuse (the side opposite the 90° angle) equals the sum of the squares on the other two sides. To find the hypotenuse, add the squares of the legs and take the square root; to find a leg, subtract the square of the other leg from the square of the hypotenuse and take the root.
Once all three sides are known, the acute angles come from trigonometry: α = arctan(a ÷ b) and β = 90° − α. The area is half the product of the two legs, because the legs are perpendicular and act as base and height.
- a, b
- The two legs that meet at the right angle
- c
- The hypotenuse, the longest side
- α, β
- The acute angles opposite a and b
Worked examples
Legs of 5 and 12
c = √(25 + 144) = √169 = 13, a classic Pythagorean triple. The area is ½ × 5 × 12 = 30 and the smaller angle is arctan(5/12) ≈ 22.62°.
Hypotenuse 10, leg 6 — find the other leg
a = √(100 − 36) = √64 = 8. This is the 3-4-5 triangle doubled, so the angles are about 53.13° and 36.87°.
A 25 ft ladder with its foot 7 ft from the wall
The ladder is the hypotenuse. The height it reaches is √(625 − 49) = √576 = 24 ft up the wall.
Frequently asked questions
What is the Pythagorean theorem?+
It states that in any right triangle the square of the hypotenuse equals the sum of the squares of the two legs: a² + b² = c². It only holds when one angle is exactly 90°.
How do you find the hypotenuse?+
Square both legs, add them and take the square root. For legs of 6 and 8: 36 + 64 = 100, and √100 = 10.
How do you find a missing leg?+
Subtract the square of the known leg from the square of the hypotenuse, then take the square root. For c = 13 and b = 12: √(169 − 144) = √25 = 5.
What are Pythagorean triples?+
Sets of whole numbers that satisfy a² + b² = c², such as 3-4-5, 5-12-13, 8-15-17 and 7-24-25. Any multiple of a triple (like 6-8-10) is also a triple.
How do builders use the 3-4-5 rule?+
To check a corner is square, mark 3 units along one side and 4 along the other. If the diagonal between the marks measures exactly 5 units, the corner is 90°. Multiples like 6-8-10 work for bigger layouts.