What Is the Pythagorean Theorem?
The Pythagorean theorem states that in a right triangle, the sum of the squares of the two legs equals the square of the hypotenuse: a² + b² = c². The hypotenuse is the side opposite the 90° angle and is always the longest side.
The theorem is traditionally associated with Pythagoras, although the relationship was known in earlier mathematical traditions. Use the calculator above to find a missing side or check whether three side lengths form a right triangle.
Pythagorean Theorem Formulas
Basic equation: a² + b² = c²
Find the hypotenuse: c = √(a² + b²)
Find side a: a = √(c² − b²)
Find side b: b = √(c² − a²)
How to Find a Missing Side
- Identify the hypotenuse, which is opposite the right angle.
- Determine which side is missing.
- Use a² + b² = c².
- Rearrange the equation when solving for a leg.
- Substitute the known side lengths.
- Square the known values.
- Add or subtract as required.
- Take the square root and round only the final decimal answer.
Worked Examples
Example 1: Find the Hypotenuse
A right triangle has legs of 9 cm and 12 cm.
c = √(9² + 12²)
c = √(81 + 144)
c = √225
c = 15 cm
Example 2: Find a Missing Leg
A right triangle has a hypotenuse of 13 m and one leg of 5 m.
b = √(13² − 5²)
b = √(169 − 25)
b = √144
b = 12 m
How to Check if a Triangle Is Right
When all three side lengths are known, sort them so the longest side is c. Then compare a² + b² with c². If the two values are equal, the triangle is a right triangle. For example, 3² + 4² = 5² because 9 + 16 = 25.
Common Pythagorean Triples
| Triple | Check |
|---|---|
| 3-4-5 | 9 + 16 = 25 |
| 5-12-13 | 25 + 144 = 169 |
| 8-15-17 | 64 + 225 = 289 |
| 7-24-25 | 49 + 576 = 625 |
Exact and Decimal Answers
Some right-triangle calculations produce perfect square roots, such as √25 = 5. Others produce irrational values, such as √13. The calculator shows an exact radical when it can be simplified cleanly and also provides a decimal approximation.
Common Pythagorean Theorem Mistakes
- Using the formula on a non-right triangle: The theorem applies only to triangles with a 90° angle.
- Misidentifying the hypotenuse: It is opposite the right angle and is always the longest side.
- Forgetting the square root: a² + b² gives c², not c.
- Subtracting in the wrong direction: When finding a leg, calculate hypotenuse² minus known-leg².
- Rounding too early: Keep the exact radical through the intermediate steps and round only the final decimal value.
How to Verify Your Answer
Square the calculated side and substitute all three sides back into a² + b² = c². Both sides of the equation should match, allowing for a very small rounding difference if a decimal approximation is used.
Pythagorean Theorem and the Distance Formula
The distance formula is the Pythagorean theorem applied to coordinate geometry. The horizontal and vertical coordinate differences form the two legs of a right triangle, while the straight-line distance is the hypotenuse. Use the Distance Between Points Calculator for coordinate-based distance problems.
Once all triangle sides are known, you can also use the Perimeter Calculator. For another triangle measurement, try the Triangle Height Calculator.
Frequently Asked Questions
No. It applies only to right triangles, which contain one 90° angle.
The hypotenuse is the side opposite the 90° angle and is always the longest side of a right triangle.
Use a = √(c² − b²) or b = √(c² − a²), where c is the hypotenuse.
Put the largest side in the c position, then test whether a² + b² = c². The calculator's Check Right Triangle mode does this automatically.
A Pythagorean triple is a set of three positive integers that satisfies a² + b² = c², such as 3-4-5 or 5-12-13.
Yes. The horizontal and vertical coordinate differences act as the legs of a right triangle, and the straight-line distance is the hypotenuse.
How we calculate
Utilixea applies a² + b² = c² to right triangles. Hypotenuse calculations use √(a² + b²), while missing-leg calculations use √(c² − known leg²). Exact radical form is preserved where possible, and a separate decimal approximation is shown for convenience.
About this calculator: Maintained by Utilixea. Calculator formulas, validation rules and test cases are reviewed whenever this tool is updated.
Last reviewed: July 22, 2026
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