Pythagorean Theorem Calculator
Calculate the hypotenuse of a right triangle instantly by entering its two legs. Ideal for construction, carpentry, and geometry homework where you need the diagonal or missing side length.
About this calculator
The Pythagorean Theorem states that in any right triangle, the square of the hypotenuse equals the sum of the squares of the other two sides. The formula is c = √(a² + b²), where a and b are the two legs and c is the hypotenuse. This relationship, proven by ancient Greek mathematicians, holds true for every right-angled triangle without exception. It is foundational in geometry, physics, and engineering — used any time you need to find a straight-line distance between two points. The theorem also works in reverse: if a² + b² = c², the triangle must be a right triangle, which is handy for verifying square corners in construction.
How to use
Suppose you are building a ramp and need to know its length. The horizontal run (Side A) is 6 units and the vertical rise (Side B) is 8 units. Plug these into the formula: c = √(6² + 8²) = √(36 + 64) = √100 = 10 units. Enter 6 in the Side A field and 8 in the Side B field, and the calculator returns 10 units immediately. This means your ramp board must be exactly 10 units long.
Frequently asked questions
How does the Pythagorean Theorem work for non-integer side lengths?
The formula c = √(a² + b²) works for any real numbers, not just whole numbers. For example, if Side A is 2.5 and Side B is 3.7, you compute √(6.25 + 13.69) = √19.94 ≈ 4.47 units. Most real-world measurements involve decimals, so the calculator handles them automatically. There is no requirement that the sides form a so-called Pythagorean triple like 3-4-5.
What is the Pythagorean Theorem used for in everyday life?
It is used wherever a right angle and two known distances appear: laying out square foundations in construction, calculating screen diagonal sizes, finding shortest walking routes across city blocks, and measuring cable lengths for rooftop antennas. Surveyors use it to compute distances across terrain, and GPS software uses it in distance calculations. Essentially any problem involving a right triangle in the real world calls on this theorem.
Can the Pythagorean Theorem be used to find a leg instead of the hypotenuse?
Yes — rearranging the formula gives a = √(c² − b²) when you know the hypotenuse and one leg. For instance, if the hypotenuse is 13 units and one leg is 5 units, the missing leg is √(169 − 25) = √144 = 12 units. This is equally common in practice, such as finding the height of a ladder's reach when you know the ladder length and its distance from the wall. Many calculators, including this one, can be adapted to solve for any one of the three sides.