Pythagorean Theorem Calculator

Solve for the hypotenuse or any missing leg of a right-angled triangle using the standard formula.

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Provide any two known side lengths (legs a and b, or a leg and the hypotenuse c) below to calculate the remaining side length automatically.

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Step by Step Calculation

a²+b²

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A Complete Guide to the Pythagorean Theorem

The Pythagorean theorem is one of the most famous and widely applied rules in geometry, establishing a fundamental relationship between the sides of a right-angled triangle. A right triangle is any three-sided shape containing a single ninety-degree angle. The theorem states that if you draw squares on each of the three sides of a right triangle, the total area of the two smaller squares will exactly equal the area of the largest square. This simple rule lets you solve physical distances in two dimensions and forms the basis for advanced geometry and trigonometry.

The Core Formula

The theorem is written as the algebraic equation: a² + b² = c². In this formula, letters a and b represent the two shorter sides (known as the legs) that meet at the right angle. The letter c represents the longest side (known as the hypotenuse), which is always located directly opposite the right angle.

To find the hypotenuse, you square both legs, add the sums, and find the square root of the result. To find a missing leg, you subtract the squared leg from the squared hypotenuse before extracting the root. To resolve square roots directly, check out our finding square root results tool. To see the flat surface enclosed by the triangle, use our flat shapes area solver.

Practical Uses of Side Lengths

Building and Construction: Carpenters and masons use the theorem to ensure corners are perfectly square by measuring diagonal lengths.
Navigation and Maps: Finding the direct line distance between two positions on a grid relies on this math. You can check coordinates with our coordinate distance mapping tool.
Measuring Screen Sizes: Televisions and smartphones are marketed by their diagonal screen measurements, which represent the hypotenuse of the display.
General Trigonometry: Solving angles in three-sided shapes begins with right-angle rules. You can analyze full profiles using our three-sided right-angled shape helper.

Pythagorean Triples

A Pythagorean triple consists of three positive whole numbers that perfectly fit the formula without any decimal remainders. The most famous example is the 3-4-5 set, where 3² + 4² = 9 + 16 = 25, which is exactly 5².

Other common triples include 5-12-13 and 8-15-17. Builders utilize these exact whole-number ratios to double-check their alignments on site. If you need to make quick additions or divisions for your ratios, use our standard daily math tools.

Example of Leaning a Ladder

Imagine you need to place a ladder against a vertical wall. The ladder is 10 feet long, and the base must sit 6 feet away from the wall for safety.

Here, the ladder is the hypotenuse (c = 10) and the ground distance is one leg (a = 6). We need to find the wall height (b). Setting up the equation: 6² + b² = 10², which simplifies to 36 + b² = 100. Subtracting 36 from both sides: b² = 64. Taking the square root reveals the ladder will reach exactly 8 feet up the wall. This simple example highlights how the formula resolves daily safety questions.