Pythagorean Theorem Calculator

What Is the Pythagorean Theorem?

The Pythagorean theorem describes the relationship between the three sides of a right triangle. In every right triangle, the square of the hypotenuse equals the sum of the squares of the two legs. Written as a formula, that is a² + b² = c², where c is always the side opposite the right angle and the longest side of the triangle.

This calculator solves that equation for any missing value. Enter the two sides you know and it returns the third, along with the area, perimeter, and step-by-step work. It is useful for students checking homework, carpenters squaring corners, and anyone estimating diagonal distances.

How to Find the Hypotenuse

When you know both legs, the hypotenuse is found by adding their squares and taking the square root. For a triangle with legs 3 and 4, the calculation is 3² + 4² = 9 + 16 = 25, and the square root of 25 is 5. The hypotenuse is therefore 5 units long.

This mode is the one most people picture first. It answers questions like "How long is the diagonal of a rectangle?" or "What is the straight-line distance between two points?" Because the hypotenuse must be the longest side, the result will always be larger than either leg.

How to Find a Missing Leg

When you know one leg and the hypotenuse, rearrange the formula to subtract the known square from the hypotenuse square, then take the square root. For example, if the hypotenuse is 13 and one leg is 12, the missing leg is √(13² - 12²) = √(169 - 144) = √25 = 5.

The calculator automatically rejects impossible combinations. If the hypotenuse is not longer than the known leg, the triangle cannot exist and the tool shows a clear error instead of a meaningless result.

Common Pythagorean Triples

Pythagorean triples are sets of whole numbers that satisfy a² + b² = c². Recognising them saves time on common problems. The table below lists the most frequently used triples.

Any multiple of a triple is also a triple. Scaling 3-4-5 by 2 gives 6-8-10, which still satisfies the theorem.

Real-World Uses of the Pythagorean Theorem

The theorem appears far beyond the classroom. Construction workers use it to make sure corners are square. TV and monitor sizes are measured diagonally, which is the hypotenuse of the screen width and height. Ladder safety guides use it to find how high a ladder reaches when leaned against a wall. GPS systems use the same idea in three dimensions to calculate distances on the earth's surface.

In design and manufacturing, the theorem helps convert between horizontal and sloped measurements. Roof pitch, ramp length, and staircase diagonal all rely on the same right-triangle relationship.

Pythagorean Theorem FAQ

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