Plot No. 61 · Education
Pythagorean Theorem Calculator
Find the missing side of a right triangle using the Pythagorean theorem, a squared plus b squared equals c squared. Enter the two legs to get the hypotenuse, or enter the hypotenuse and one leg to get the other leg, with the equation shown for each answer.
c = sqrt(3^2 + 4^2) = sqrt(25) = 5.0000
The Pythagorean theorem only applies to right-angled triangles, where a squared plus b squared equals c squared.
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How the Pythagorean theorem is applied
The Pythagorean theorem says that in a right-angled triangle the square of the hypotenuse equals the sum of the squares of the two shorter sides, written a squared plus b squared equals c squared. This calculator works in two directions. In hypotenuse mode it squares both legs, adds them and takes the square root to find c. In leg mode it subtracts the square of the known leg from the square of the hypotenuse and takes the square root to find the missing leg.
Because the hypotenuse is always the longest side, leg mode requires the entered hypotenuse to be longer than the known leg; otherwise the squared difference would be negative and no real triangle exists, so the calculator shows a guard message. Each answer is given to four decimal places along with the exact equation used.
How to use the Pythagorean theorem calculator
- Pick whether you want to solve for the hypotenuse or for a missing leg.
- In hypotenuse mode enter the two legs; in leg mode enter the hypotenuse and the leg you already know.
- Read the missing side and the equation that produced it as you type.
Worked examples
Finding the hypotenuse
Inputs: Legs of 3 and 4
Result: c equals the square root of 9 plus 16, which is the square root of 25, so the hypotenuse is exactly 5.
Finding a missing leg
Inputs: Hypotenuse 13, known leg 5
Result: The leg equals the square root of 169 minus 25, which is the square root of 144, so the missing leg is exactly 12.
Limitations and common mistakes
Edge cases and limitations
- It only applies to right-angled triangles; it cannot solve triangles without a 90 degree angle.
- In leg mode the hypotenuse must be strictly longer than the known leg, otherwise no real triangle exists and a guard message appears.
- Answers are rounded to four decimal places, so irrational results like the square root of 2 are shown as approximations.
Common mistakes
- Treating the longest given side as a leg when finding a missing leg; the hypotenuse must be the value you subtract from.
- Adding the sides instead of their squares; 3 plus 4 is 7, but the hypotenuse of a 3 by 4 right triangle is 5, not 7.
Frequently asked questions
What is the Pythagorean theorem?
In any right-angled triangle, the square of the hypotenuse (the longest side, opposite the right angle) equals the sum of the squares of the other two sides. Written as a formula, that is a squared plus b squared equals c squared.
How do you find the hypotenuse?
Square both legs, add them, then take the square root. For legs of 3 and 4, you get 9 plus 16, which is 25, and the square root of 25 is 5, so the hypotenuse is 5.
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. The hypotenuse must be longer than the known leg, otherwise the triangle cannot exist. For a hypotenuse of 13 and a leg of 5, the other leg is the square root of 144, which is 12.
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