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Pythagorean Theorem Calculator

Use a²+b²=c² to find a right triangle's hypotenuse or leg

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How to use

  1. 1.Choose whether to find the hypotenuse (from two legs) or a missing leg (from the hypotenuse and one leg).
  2. 2.Type the two lengths you already know into the input boxes.
  3. 3.Read the missing side instantly, along with the area, perimeter and both acute angles.

About Pythagorean Theorem Calculator

The Pythagorean theorem is one of the oldest and most useful rules in geometry. It states that in any right triangle — a triangle with one 90° angle — the square of the hypotenuse equals the sum of the squares of the other two sides: a² + b² = c². Here a and b are the two legs that meet at the right angle, and c is the hypotenuse, the longest side sitting opposite that angle. This pythagorean theorem calculator lets you type in the sides you know and instantly solves for the one you don't.

The formula a² + b² = c² only works for right triangles, and that single condition is what makes it so powerful. It links the three sides together so tightly that knowing any two of them fixes the third. If you know both legs, the hypotenuse is c = √(a² + b²). If you know the hypotenuse and one leg, the other leg is b = √(c² − a²). Because the hypotenuse is always the longest side, c must be greater than either leg — this right triangle calculator checks that for you and flags impossible inputs where the hypotenuse would be shorter than a leg.

The tool does more than return a single side. As soon as all three sides are known it also reports the area (½ · a · b), the perimeter (a + b + c), and the two acute angles, which it works out with the arctangent and which always add up to 90°. Every result updates the moment you change an input, so you can experiment with different values and watch the triangle respond in real time. Numbers are rounded to a readable number of decimal places while keeping enough precision to verify exact triples, and the working is shown as a formula line such as c = √(a² + b²) so you can follow each step rather than trusting a black box. That makes it a quick right triangle solver for homework, drafting, and everyday measuring, not just a hypotenuse calculator.

Some side triples are so clean they are worth memorising. The classic 3-4-5 triangle satisfies 3² + 4² = 5² (9 + 16 = 25), and any multiple of it — 6-8-10, 9-12-15 — is also a right triangle. Other well-known Pythagorean triples include 5-12-13 and 8-15-17. Seeing these come out exactly is a good way to confirm the calculator is behaving, and they show up constantly in textbooks and real construction.

Real-world uses are everywhere. Builders use the theorem to square up walls and check that corners are true right angles. Navigators and surveyors use it to turn north-south and east-west distances into a straight-line distance. It is the reason a TV or monitor 'size' — the diagonal — can be found from its width and height, and it underlies the distance formula used in coordinate geometry and game development. Enter your known lengths above to find the missing side, the area, the perimeter, and the angles, all computed privately in your browser with nothing uploaded.

Frequently asked questions

What is the Pythagorean theorem?
The Pythagorean theorem says that in a right triangle the square of the hypotenuse equals the sum of the squares of the other two sides: a² + b² = c². Here a and b are the legs that form the right angle and c is the hypotenuse opposite it. It was recorded by Euclid as Proposition 47 in Book I of the Elements.
How do I find the hypotenuse?
Square both legs, add them, then take the square root: c = √(a² + b²). For example, with legs of 3 and 4 the hypotenuse is √(9 + 16) = √25 = 5. Select 'Find hypotenuse', enter the two legs, and the calculator does this instantly.
How do I find a missing leg?
Rearrange the formula to b = √(c² − a²): square the hypotenuse, subtract the square of the known leg, then take the square root. With a hypotenuse of 13 and a leg of 5, the other leg is √(169 − 25) = √144 = 12. The hypotenuse must be longer than the leg, or no right triangle exists.
Does the Pythagorean theorem work for non-right triangles?
No. a² + b² = c² holds only for right triangles, those with exactly one 90° angle. For other triangles you need the Law of Cosines, c² = a² + b² − 2ab·cos(C), which reduces to the Pythagorean theorem when angle C is 90° and cos(C) is 0.
What is a 3-4-5 triangle?
A 3-4-5 triangle is a right triangle whose sides are in the ratio 3:4:5, since 3² + 4² = 9 + 16 = 25 = 5². It is the simplest Pythagorean triple, and any multiple such as 6-8-10 or 9-12-15 is also a right triangle. Builders use it to mark out perfect right angles.

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