Pythagorean Theorem Calculator
This Pythagorean theorem calculator finds the missing side of a right triangle in seconds. Choose whether you want to work out the hypotenuse — the longest side, opposite the right angle — from the two legs, or find one leg when you already know the hypotenuse and the other leg. Just type the two known lengths and the result updates instantly, no button needed. The tool is built on the classic relationship a² + b² = c², one of the most useful formulas in all of geometry. It is perfect for students checking homework, carpenters squaring a frame, surveyors measuring distances and anyone who needs a quick, reliable answer for a right-angled triangle.
Formula: a² + b² = c². Only valid for right triangles (those with a 90° angle).
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How it works
The Pythagorean theorem states that in a right triangle the square of the hypotenuse equals the sum of the squares of the two legs: a² + b² = c². To find the hypotenuse you add the squares of the legs and take the square root: c = √(a² + b²). To find a missing leg you rearrange the formula and subtract instead: b = √(c² − a²). Take the classic 3-4-5 triangle: with legs a = 3 and b = 4, the hypotenuse is c = √(3² + 4²) = √(9 + 16) = √25 = 5. Working backwards, if c = 5 and a = 3, then b = √(5² − 3²) = √(25 − 9) = √16 = 4. Remember that the hypotenuse is always the longest side, so a leg is only valid when the hypotenuse is greater than the known leg.
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Preguntas frecuentes
- What is the Pythagorean theorem?
- The Pythagorean theorem is a rule about right triangles that states a² + b² = c², where a and b are the two legs (the sides forming the right angle) and c is the hypotenuse (the side opposite the right angle). It lets you find any one side when you know the other two.
- How do I find the hypotenuse?
- Square both legs, add them together and take the square root of the sum: c = √(a² + b²). For example, with legs of 3 and 4, the hypotenuse is √(9 + 16) = √25 = 5. Select the "Find the hypotenuse" mode and enter the two legs to get the answer instantly.
- How do I find a missing leg?
- Rearrange the formula to subtract: b = √(c² − a²). You need the hypotenuse and one leg. For instance, if the hypotenuse is 5 and one leg is 3, the other leg is √(25 − 9) = √16 = 4. Use the "Find a leg" mode and enter the hypotenuse and the known leg.
- Does the theorem work for any triangle?
- No. The Pythagorean theorem only works for right triangles — triangles that contain exactly one 90° angle. For triangles without a right angle you would need other tools such as the law of cosines or the law of sines.
- Why must the hypotenuse be longer than each leg?
- Because the hypotenuse is opposite the largest angle (the 90° angle), it is always the longest side. When finding a leg, the hypotenuse you enter must be greater than the known leg; otherwise the numbers do not describe a valid right triangle and no real answer exists.