Pythagorean Theorem Calculator — a² + b² = c²
Solve for any missing side of a right triangle using the Pythagorean theorem. Includes a Pythagorean triple checker and common triples reference. See also Right Triangle Calculator and Trigonometry Calculator.
Solve for a Missing Side
Pythagorean Triple Checker
Enter three side lengths to check if they form a right triangle (satisfy a² + b² = c²).
How to Use the Pythagorean Theorem
The Pythagorean theorem states that in a right triangle, the square of the hypotenuse (the side opposite the right angle) equals the sum of the squares of the other two sides: a² + b² = c². To find a missing side, select what you want to solve for, enter the two known values, and click Calculate. The calculator shows the missing side, all three sides, a step-by-step verification, plus the area and perimeter.
Pythagorean Theorem Formula
a² + b² = c²
Solve for c: c = √(a² + b²)
Solve for a: a = √(c² − b²)
Solve for b: b = √(c² − a²)
Area = ½ × a × b
Perimeter = a + b + c
Example Calculation
Find the hypotenuse when a = 3 and b = 4:
c² = a² + b²
c² = 3² + 4² = 9 + 16 = 25
c = √25 = 5
Find side a when b = 12 and c = 13:
a² = c² − b²
a² = 13² − 12² = 169 − 144 = 25
a = √25 = 5
Common Pythagorean Triples
| a | b | c | a² + b² | c² | Type |
|---|---|---|---|---|---|
| 3 | 4 | 5 | 25 | 25 | Primitive |
| 5 | 12 | 13 | 169 | 169 | Primitive |
| 8 | 15 | 17 | 289 | 289 | Primitive |
| 7 | 24 | 25 | 625 | 625 | Primitive |
| 9 | 40 | 41 | 1681 | 1681 | Primitive |
| 11 | 60 | 61 | 3721 | 3721 | Primitive |
| 12 | 35 | 37 | 1369 | 1369 | Primitive |
| 13 | 84 | 85 | 7225 | 7225 | Primitive |
| 20 | 21 | 29 | 841 | 841 | Primitive |
| 28 | 45 | 53 | 2809 | 2809 | Primitive |
| 6 | 8 | 10 | 100 | 100 | 3, 4, 5 × 2 |
| 9 | 12 | 15 | 225 | 225 | 3, 4, 5 × 3 |
| 10 | 24 | 26 | 676 | 676 | 5, 12, 13 × 2 |
| 15 | 20 | 25 | 625 | 625 | 3, 4, 5 × 5 |
| 20 | 48 | 52 | 2704 | 2704 | 5, 12, 13 × 4 |
Frequently Asked Questions
What is the Pythagorean theorem?
The Pythagorean theorem states that in any right triangle, the square of the hypotenuse equals the sum of the squares of the other two sides: a² + b² = c². It is one of the most fundamental theorems in mathematics, attributed to the ancient Greek mathematician Pythagoras.
What is a Pythagorean triple?
A Pythagorean triple is a set of three positive integers (a, b, c) that satisfy a² + b² = c². The most famous is (3, 4, 5). A primitive triple has no common factor greater than 1. Any multiple of a Pythagorean triple is also a triple — for example, (6, 8, 10) = 2 × (3, 4, 5).
Does the Pythagorean theorem work for all triangles?
No — it only applies to right triangles (triangles with one 90° angle). For non-right triangles, use the law of cosines: c² = a² + b² − 2ab·cos(C), which generalizes the Pythagorean theorem. When C = 90°, cos(C) = 0 and it reduces to a² + b² = c².
How can I tell if a triangle is right, acute, or obtuse?
Given three sides where c is the longest: if a² + b² = c², it is a right triangle. If a² + b² > c², it is acute (all angles less than 90°). If a² + b² < c², it is obtuse (one angle greater than 90°). The Pythagorean triple checker above performs this test.
What are real-world applications of the Pythagorean theorem?
The Pythagorean theorem is used in construction (ensuring walls are square), navigation (calculating straight-line distances), computer graphics (distance between points), surveying, architecture, and physics. Any time you need to find a distance in 2D or 3D space, the Pythagorean theorem is involved.