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Pythagorean Theorem Calculator
Find a missing leg or hypotenuse of a right triangle, or check three sides with the converse. See a² + b² = c² with your numbers filled in.
Which side is unknown?
One side that forms the right angle
The other side that forms the right angle
c = 5
Formula
a² + b² = c²
With your numbers
3² + 4² = c²
9 + 16 = c²
c² = 25
c = √25 = 5
Result
This Pythagorean theorem calculator finds the missing leg or hypotenuse of a right triangle and checks whether three lengths form a right triangle. Mark the unknown side, enter the two you know, and the page writes a² + b² = c² with those numbers before it shows the missing length. When the square root is irrational, a perfect-square factor is taken out of the radical and a decimal is shown beside the simplified root. The check names the hypotenuse when the two smaller squares add up to the largest square.
Legs, hypotenuse, and a² + b² = c²
A right triangle has one right angle. The legs, a and b, are the sides that form that angle. The hypotenuse, c, is the side across from the right angle, and it is the longest of the three. The Pythagorean theorem says the square of the hypotenuse equals the sum of the squares of the legs: a² + b² = c². A 3-4-5 triangle is the usual classroom check, because 9 + 16 = 25.
Finding the hypotenuse means adding the squares of the legs and taking the square root. Finding a leg means subtracting: square the hypotenuse, subtract the square of the known leg, and take the square root. For a hypotenuse of 10 and a leg of 6, that is 100 − 36 = 64, and the square root of 64 is 8. The subtraction stays positive only when the hypotenuse is longer than the known leg. Equal lengths or a longer leg leave 0 or a negative number under the root, and those lengths do not make a triangle.
Checking three sides with the converse
The converse of the Pythagorean theorem is the test you use when you have all three lengths and want to know whether the triangle is right. Square the two shorter sides and the longest side. If the two smaller squares add up to the largest square, the triangle is a right triangle and the longest side is the hypotenuse. Sides 6, 8, and 10 satisfy it, since 36 + 64 = 100. Sides 2, 3, and 4 do not: 4 + 9 = 13, and 4² is 16. An irrational missing side, such as the hypotenuse when both legs are 1, stays in radical form. √2 is already simplified, and a decimal sits beside it.
Frequently Asked Questions
What is the Pythagorean theorem?
In a right triangle, the square of the hypotenuse equals the sum of the squares of the two legs. With legs a and b and hypotenuse c, that relationship is a² + b² = c². The hypotenuse is the side opposite the right angle, and it is the longest side.
How do you find a missing hypotenuse?
Square each known leg, add the squares, and take the square root. Legs 3 and 4 give 3² + 4² = 9 + 16 = 25, so the hypotenuse is √25, which is 5. Legs 5 and 12 give a hypotenuse of 13.
How do you find a missing leg?
Square the hypotenuse, subtract the square of the known leg, and take the square root: a² = c² − b². A hypotenuse of 10 and a leg of 6 give 100 − 36 = 64, so the missing leg is 8. If the hypotenuse is shorter than the known leg, or the same length, the number under the root is zero or negative and there is no real triangle.
What is the converse of the Pythagorean theorem?
If the two smaller squares add up to the largest square, the triangle is a right triangle, and the longest side is the hypotenuse. Sides 6, 8, and 10 pass, because 36 + 64 = 100. Sides 2, 3, and 4 fail, because 4 + 9 = 13 and the square of the longest side is 16.
Why is the answer sometimes left as a square root?
The length is irrational when the number under the root has no whole-number square root. The calculator simplifies perfect-square factors, so √8 is written 2√2, and √2 stays √2, with a decimal beside the radical. Enter every side in the same unit, such as centimeters or inches.