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Pythagorean Theorem Calculator

Fill in any two sides of a right triangle — the two legs, or one leg and the hypotenuse — and this tool solves for the missing side using a² + b² = c².

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📌 Enter Two Sides

Leave the unknown side blank — fill in any 2 of the 3 fields.

📊 Result

Fill in any two sides to solve for the third
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Disclaimer: This tool performs standard mathematical calculations for educational and reference purposes. Always double-check results independently for graded coursework, exams, or safety-critical applications.

Understanding the Pythagorean Theorem

The Pythagorean theorem describes a fixed relationship between the three sides of any right triangle — a triangle containing one 90° angle. It states that the square of the hypotenuse (the longest side, opposite the right angle) equals the sum of the squares of the other two sides, called the legs.

The Formula

a² + b² = c², where a and b are the two legs and c is the hypotenuse. Rearranged, this gives three versions depending on which side is missing: c = √(a² + b²), a = √(c² − b²), or b = √(c² − a²).

Worked Example

A right triangle has legs of 3 units and 4 units. Find the hypotenuse.

  1. Step 1: a² + b² = 3² + 4² = 9 + 16 = 25
  2. Step 2: c = √25 = 5 units

Because 3, 4, and 5 are all whole numbers, this is a well-known Pythagorean triple. Any multiple of it — 6-8-10, 9-12-15, and so on — also forms a valid right triangle with whole-number sides.

Frequently Asked Questions

The relationship a² + b² = c² comes directly from the geometry of a 90° corner between the two legs. If the angle between those two sides isn't exactly 90°, the squared relationship no longer holds, and you'd need a more general formula like the law of cosines instead.
The hypotenuse is always the longest side of a right triangle, and it's always the side directly opposite the 90° angle. The two shorter sides forming the right angle itself are called the legs.
A 3-4-5 triangle is a right triangle whose three sides are in the exact ratio 3:4:5 (or any multiple of it, like 6-8-10 or 9-12-15). These are called Pythagorean triples because all three side lengths come out as whole numbers, which makes them handy for construction and quick mental checks.
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