Pythagorean Theorem Calculator
Calculate the missing side of a right triangle using the Pythagorean theorem.
Understanding Pythagorean Theorem Calculator
The Pythagorean theorem describes the relationship between the three side lengths of a right triangle. If a and b are the two legs and c is the hypotenuse, the fundamental relationship is a² + b² = c².
The hypotenuse is the side opposite the right angle and is always the longest side of a right triangle. The theorem allows a missing side to be calculated when the other two relevant side lengths are known.
Using this calculator with your own numbers
Pythagorean Theorem Calculator is intended to show the mathematical relationship behind the answer, not just return a number. Enter the quantities requested by the selected method—such as Side A, Side B, Find, Hypotenuse (c), if needed—and keep their units or mathematical meaning consistent. Where this calculator offers different shapes, methods, or forms, choose the one that matches the problem before entering the values.
When the result is an estimate
For a useful check, look at the units and the scale of the answer before accepting it. A formula can be evaluated correctly while an input is still inappropriate for the problem. Repeating the calculation with a simple test value or checking the result against the expected direction of change can help identify an input mistake.
Pythagorean Theorem Calculator
The Pythagorean theorem describes the relationship between the three side lengths of a right triangle. If a and b are the two legs and c is the hypotenuse, the fundamental relationship is a² + b² = c².
The hypotenuse is the side opposite the right angle and is always the longest side of a right triangle. The theorem allows a missing side to be calculated when the other two relevant side lengths are known.
The Pythagorean Theorem Formula
Here, a and b represent the two perpendicular legs of the right triangle, while c represents the hypotenuse.
Finding the Hypotenuse
When both legs are known, the hypotenuse can be calculated by taking the square root of the sum of their squared lengths.
For example, if the two legs are 3 and 4 units, then the hypotenuse is 5 units because 3² + 4² = 5².
Finding Side A
If the hypotenuse and side B are known, the missing side A can be found by rearranging the Pythagorean theorem.
The hypotenuse must always be greater than side B for a positive, valid value of A to exist.
Finding Side B
If the hypotenuse and side A are known, side B can be calculated using the corresponding rearrangement.
The hypotenuse must be greater than side A. Otherwise, the supplied measurements cannot represent the required right triangle.
Why the Theorem Works
Geometrically, the Pythagorean theorem states that the area of the square constructed on the hypotenuse equals the combined areas of the squares constructed on the two legs. This relationship is the basis for many calculations in Euclidean geometry.
Connection to Coordinate Geometry
The distance formula used to find the straight-line distance between two points is a direct application of the Pythagorean theorem. The horizontal and vertical differences between the points form the two legs of a right triangle.
Important Conditions
The Pythagorean theorem applies specifically to right triangles. It should not be used directly for arbitrary triangles that do not contain a 90° angle.
When calculating a missing leg, the hypotenuse must be longer than the known leg. The calculator checks these conditions before producing a result.
How to Use the Calculator
Enter the known side lengths and select which side you want to find. When calculating the hypotenuse, enter sides A and B. When calculating side A or B, enter the other leg and the hypotenuse.
The result shows the missing side, the values used, the applicable formula, the calculation steps, and a verification of the Pythagorean relationship.
Disclaimer
This calculator is intended for informational purposes only. While we strive for accuracy, we cannot guarantee that every result is correct for every situation.