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  4. Pythagorean Theorem Calculator

Pythagorean Theorem Calculator

Solve a² + b² = c² given any two sides of a right triangle. See squares on each side and stepped solutions.

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How it works

In a right triangle the area of the square on the hypotenuse equals the sum of the squares on the legs.

a² + b² = c²

When to use

Use this tool when you need to: leave c blank with a=3, b=4 to get c=5. Try 5-12-13 to verify a triple.

How to use this tool

  1. Enter your values in the fields above.
  2. Review the formula and any mode options for your problem.
  3. Read the result and the step-by-step solution.

Leave c blank with a=3, b=4 to get c=5. Try 5-12-13 to verify a triple.

Frequently asked questions

What is the Pythagorean Theorem Calculator?
In a right triangle the area of the square on the hypotenuse equals the sum of the squares on the legs.
How do I use the Pythagorean Theorem Calculator?
Enter the required values, then calculate. The Pythagorean Theorem Calculator shows results, formulas, and step-by-step work when available. No account is required.
What formula does the Pythagorean Theorem Calculator use?
a² + b² = c²

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Pythagorean theorem calculator

Provide any two of a, b, and hypotenuse c to solve a² + b² = c². See side lengths, squares on each side, and a diagram linking the areas.

Leave exactly one field blank for the unknown side (or enter a known triple to verify). c is always the longest side — the hypotenuse.

Diagram of the right triangle and squares on each side appears after solve.

a² + b² = c² · c = √(a²+b²) · a = √(c²−b²) · b = √(c²−a²)

Try an example

How it works

What is the Pythagorean theorem?

In any right triangle (one 90° angle), if c is the hypotenuse and a, b are the legs, then a² + b² = c². Geometrically: the area of the square built on the longest side equals the sum of the areas of the squares built on the other two sides.

Solving for a missing side

Given a = 3 and b = 4: c = √(a² + b²) = √(9+16) = √25 = 5. To find a leg: a = √(c² − b²) or b = √(c² − a²). The unknown side must leave a positive radicand — so c must exceed each known leg.

Algebraic square packing

One classic proof arranges four copies of the right triangle around a square of side c inside a larger square of side a + b. Comparing areas: (a+b)² = c² + 4 · (½ab) leads to a² + b² = c². A second packing uses an inner square of side |b − a| with the four triangles filling a square of side c and yields the same identity.

Link to the law of cosines

The law of cosines states c² = a² + b² − 2ab cos C for any triangle. When the angle C between sides a and b is a right angle, cos 90° = 0 and the formula reduces to the Pythagorean equation.

Related geometry

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