Pythagorean Theorem Calculator

How to Use the Pythagorean Theorem Calculator

This calculator solves for any side of a right triangle using the Pythagorean theorem. Each card corresponds to one unknown: the hypotenuse (c) or either leg (a or b). Enter the two known side lengths and the missing side is computed instantly. There is no need to press a button or reload the page; results update in real time as you type.

The Pythagorean theorem is one of the oldest and most widely used results in mathematics. Attributed to the ancient Greek mathematician Pythagoras, it establishes a relationship between the three sides of every right triangle. Carpenters use it to square walls, surveyors use it to measure land, and programmers use it to calculate distances between points on a screen.

Understanding the Pythagorean Theorem

The theorem states that in any right triangle the square of the hypotenuse equals the sum of the squares of the other two sides: a² + b² = c². The hypotenuse is always the longest side and is opposite the right angle. By rearranging the equation you can isolate any side:

• c = √(a² + b²) — find the hypotenuse from two legs
• a = √(c² − b²) — find a leg from the hypotenuse and the other leg
• b = √(c² − a²) — find the remaining leg

When solving for a leg, the value under the square root must be positive. If c² is smaller than the other side squared, no valid right triangle exists with those measurements.

Pythagorean Triples

A Pythagorean triple is a set of three positive integers (a, b, c) that satisfies the theorem exactly. The most famous triple is (3, 4, 5). Others include (5, 12, 13), (8, 15, 17), and (7, 24, 25). Multiplying every element of a triple by the same factor produces another triple, so (6, 8, 10) and (9, 12, 15) also work. Builders frequently use the 3-4-5 rule to verify that corners are square without needing special equipment.

Real-World Applications

Beyond the classroom, the Pythagorean theorem appears in navigation, architecture, computer graphics, and physics. GPS systems use it (in three dimensions) to pinpoint locations. Game developers compute distances between objects with it. Electricians calculate wire runs through walls and ceilings. Whenever you need the straight-line distance between two points in a coordinate system, the Pythagorean theorem is the tool to use.

Frequently Asked Questions

What is the Pythagorean theorem?

The Pythagorean theorem states that in a right triangle the square of the hypotenuse (c) equals the sum of the squares of the other two sides: a² + b² = c². It is one of the most fundamental relationships in geometry.

How do you find the hypotenuse of a right triangle?

Square each of the two shorter sides (legs), add the results together, then take the square root. For example, if a = 3 and b = 4, then c = √(9 + 16) = √25 = 5.

How do you find a missing leg of a right triangle?

Rearrange the formula to a = √(c² − b²) or b = √(c² − a²). Subtract the square of the known leg from the square of the hypotenuse, then take the square root of the result.

Does the Pythagorean theorem work for all triangles?

No, it only applies to right triangles — triangles that have one 90-degree angle. For non-right triangles, use the Law of Cosines: c² = a² + b² − 2ab·cos(C), which generalises the Pythagorean theorem.

What are Pythagorean triples?

Pythagorean triples are sets of three positive integers that satisfy a² + b² = c². Common examples include (3, 4, 5), (5, 12, 13), (8, 15, 17), and (7, 24, 25). Any multiple of a triple is also a triple, such as (6, 8, 10).