The most useful theorem in geometry
The Pythagorean theorem connects the three sides of any right-angled triangle, and it has been used for thousands of years in building, navigation, and surveying. It states that the square of the longest side (the hypotenuse) equals the sum of the squares of the other two sides. If you know any two sides, you can always find the third.
How to use the calculator
Enter any two of the three sides — leave the one you want to find blank. The calculator works out the missing side and labels it for you. The hypotenuse is always the longest side and sits opposite the right angle.
The formula in three forms
The base equation is a² + b² = c². To find the hypotenuse, use c = √(a² + b²). To find a shorter side, rearrange to a = √(c² − b²). The famous 3-4-5 triangle demonstrates it perfectly: 3² + 4² = 9 + 16 = 25 = 5².
Where it is used in the real world
Builders use the 3-4-5 rule to check that corners are perfectly square without a set square — measure 3 units along one wall, 4 along the other, and if the diagonal is exactly 5, the corner is 90°. The theorem also underpins the distance formula in coordinate geometry, GPS positioning, and any situation where you need the straight-line distance between two points that differ in both horizontal and vertical position.