Pythagorean Theorem Calculator
Calculate any side of a right triangle from the other two. Enter two sides and get the third with the formula a² + b² = c².
Fill in two fields to calculate the third.
The Theorem
For any right triangle, the relation a² + b² = c² holds: squaring the hypotenuse c gives the same result as adding the squares of the legs a and b. The legs are the two sides that form the right angle, while the hypotenuse sits opposite that angle and is always the longest side.
The Pythagorean theorem
In a right triangle (one 90° angle), a² + b² = c², where a and b are the legs and c is the hypotenuse — the side opposite the right angle. The converse also holds: if a² + b² = c², the triangle is right-angled. Example: legs 3 and 4 give c = √(9 + 16) = √25 = 5. The relation was known to pre-Greek civilizations (Babylonian tablet Plimpton 322, Egyptian rope-stretchers) but Pythagoras of Samos (~530 BC) is credited with a general proof; more than 400 distinct proofs are documented today. Pythagorean triples are integer solutions: (3, 4, 5), (5, 12, 13), (8, 15, 17), (7, 24, 25), (20, 21, 29). A generalisation to any triangle is the law of cosines: c² = a² + b² − 2ab·cos C, which reduces to Pythagoras when C = 90°. The 2D Euclidean distance d = √((x₂−x₁)² + (y₂−y₁)²) is Pythagoras applied to coordinates.
Applications
Construction (the 3-4-5 square method to set walls and foundations at right angles), surveying, navigation, GPS triangulation, computer graphics (vector lengths and collision detection), screen-diagonal computation and any physics problem involving perpendicular components (force vectors, projectile motion).
FAQ
Does it work for any triangle? No — only right triangles. For others use the law of cosines.
Which side is the hypotenuse? The longest side, opposite the 90° angle. It is always c in the formula.
What is a Pythagorean triple? Three positive integers (a, b, c) with a² + b² = c². The smallest is (3, 4, 5); multiples like (6, 8, 10) also count.
Did Pythagoras really discover it? The relation was used centuries earlier in Babylonia and Egypt; Pythagoras (or his school) is credited with the first general proof in the Greek tradition.
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The Pythagorean theorem in practice
Few geometry relationships get as much mileage as a² + b² = c². It shows up on the school test and turns up again when you're doing a job around the house. What it does is tie together the three sides of a right triangle, and this tool finds any one of those sides once you already have the other two.
There are two paths. Give it the two legs and it finds the hypotenuse; give it one leg and the hypotenuse and it returns the missing side. That covers maths exercises and also real problems, like checking whether a corner came out square or measuring the diagonal of a room.
Nothing needs installing, since the calculation happens right in the browser. Use it as a reference for studying, teaching, or just getting through a problem that involves right angles.