Pythagorean Theorem Calculator
Find the hypotenuse of any right triangle by entering the two shorter sides. Ideal for construction, navigation, screen size comparisons, and geometry problems.
About this calculator
The Pythagorean theorem states that in any right triangle, the square of the hypotenuse (the side opposite the right angle) equals the sum of the squares of the other two sides: c = √(a² + b²). Here, a and b are the two legs and c is the hypotenuse. This relationship holds for every right triangle regardless of size and is one of the most fundamental results in Euclidean geometry. It is used to find straight-line distances in two-dimensional space — the formula is essentially the same as the Euclidean distance formula between two points. Practical applications include carpentry (checking square corners), navigation (shortest-path distances), screen diagonal measurements, and physics (vector magnitudes). The theorem extends to three dimensions as c = √(a² + b² + d²) for spatial diagonals.
How to use
Imagine you are building a ramp that spans 6 feet horizontally and rises 2.5 feet vertically. Enter Side A = 6 and Side B = 2.5. The calculator computes c = √(6² + 2.5²) = √(36 + 6.25) = √42.25 ≈ 6.50 feet. That is the length of lumber you need for the ramp surface. As a quick check: the hypotenuse must always be longer than either leg individually — 6.50 is indeed greater than both 6 and 2.5, so the result is plausible.
Frequently asked questions
How do I find a missing leg of a right triangle instead of the hypotenuse?
Rearrange the formula: if you know the hypotenuse c and one leg a, the missing leg is b = √(c² − a²). For example, if a ladder of length 10 m leans against a wall and its base is 4 m from the wall, the height it reaches is √(10² − 4²) = √(100 − 16) = √84 ≈ 9.17 m. This reverse application is just as common as finding the hypotenuse, particularly in construction and navigation problems.
What are Pythagorean triples and why are they useful?
A Pythagorean triple is a set of three positive integers (a, b, c) that satisfy a² + b² = c² exactly, with no rounding. The most famous example is (3, 4, 5): 9 + 16 = 25. Other common triples include (5, 12, 13) and (8, 15, 17). These are extremely useful in construction and carpentry because they guarantee a perfect right angle using only whole-number measurements — no calculator needed on a job site. Builders use the 3-4-5 rule constantly when squaring up foundations and walls.
Does the Pythagorean theorem work for non-right triangles?
No — the standard theorem applies only to right triangles (those containing a 90° angle). For other triangles, you need the Law of Cosines: c² = a² + b² − 2ab × cos(C), where C is the angle between sides a and b. When C = 90°, cos(90°) = 0 and the formula reduces exactly to the Pythagorean theorem, confirming it as a special case. The Law of Cosines works for any triangle, making it the more general tool for trigonometry and surveying applications.