Pythagorean Theorem Calculator
Calculate the hypotenuse or a missing leg of a right triangle using the Pythagorean theorem, a² + b² = c².
- a²9 · 36%
- b²16 · 64%
The classic picture: a square built on side a plus a square built on side b together cover exactly the same area as a square built on the hypotenuse (c² = 25).
ƒShow your work
c = √(a² + b²)
- 1c = √(3² + 4²) = 5
About the Pythagorean Theorem Calculator
The Pythagorean theorem is one of the few pieces of geometry that keeps showing up long after school: a carpenter squaring a corner with the 3-4-5 rule, someone figuring out the diagonal brace needed for a fence or shelf, a TV or monitor's diagonal size measurement, or a hiker estimating a straight-line distance from two perpendicular legs of a route. It only applies to right triangles, but a huge number of practical measuring problems can be set up as one.
This calculator solves both directions: finding the hypotenuse from the two legs, or finding a missing leg when you know the hypotenuse and the other leg — the second case comes up just as often in practice, like knowing a ladder's length and how far its base sits from a wall, and needing to know how high it reaches.
How it’s calculated
The theorem states a² + b² = c², where a and b are the two legs of a right triangle and c is the hypotenuse (the side opposite the right angle, always the longest side). To find the hypotenuse: c = √(a² + b²). To find a missing leg when you know the hypotenuse and the other leg: b = √(c² − a²) — the same equation, just rearranged.
Geometrically, the theorem says that a square built on each leg, added together, covers exactly the same area as a square built on the hypotenuse — that's the classic visual proof, and it's also exactly why the formula squares each side before adding.
Frequently asked questions
Does the Pythagorean theorem work on any triangle?
No — it only applies to right triangles, ones with a 90° angle. For any other triangle, use the law of cosines instead, which is a more general version that reduces to the Pythagorean theorem exactly when one angle is 90°.
How do I find a leg instead of the hypotenuse?
Rearrange the formula to b = √(c² − a²), using the hypotenuse and the known leg. This calculator's 'find missing leg' mode does exactly this, and it requires the hypotenuse to be the longest of the two known values, since it's always the longest side of the triangle.
What are Pythagorean triples?
Sets of three whole numbers that satisfy a² + b² = c² exactly, with no decimals involved — 3-4-5 is the most famous, along with 5-12-13 and 8-15-17. Carpenters use the 3-4-5 triple specifically to check that a corner is a true right angle without any measuring tools beyond a tape measure.
What are some real-world uses of the Pythagorean theorem?
Squaring corners in construction, finding a TV or monitor's actual screen size from width and height, calculating the shortest straight-line distance between two points that differ in both horizontal and vertical position, and figuring out how far a leaning ladder reaches up a wall.
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