Pythagorean Theorem
Missing side of a right triangle
Leave one field blank to solve for it. a² + b² = c²
About This Calculator
The Pythagorean theorem relates the three sides of any right triangle: the square of the hypotenuse equals the sum of the squares of the two legs. It is one of the oldest and most proven theorems in mathematics with applications in construction, navigation, and physics.
Formula
a^2 + b^2 = c^2 (c = hypotenuse, a and b = legs)
Hypotenuse: c = sqrt(a^2 + b^2)
Leg: a = sqrt(c^2 - b^2)
Example Calculation
Right triangle with legs a=6 and b=8
- c^2 = 6^2 + 8^2 = 36 + 64 = 100
- c = sqrt(100) = 10
Hypotenuse = 10
Common Pythagorean Triples
| a | b | c | Scale (x2) | Scale (x3) |
|---|---|---|---|---|
| 3 | 4 | 5 | 6-8-10 | 9-12-15 |
| 5 | 12 | 13 | 10-24-26 | 15-36-39 |
| 8 | 15 | 17 | 16-30-34 | 24-45-51 |
| 7 | 24 | 25 | 14-48-50 | 21-72-75 |
| 9 | 40 | 41 | 18-80-82 | 27-120-123 |
Frequently Asked Questions
What is a Pythagorean triple?
A Pythagorean triple is a set of three positive integers (a, b, c) satisfying a^2 + b^2 = c^2. The most famous is 3-4-5. Any multiple of a triple (like 6-8-10) is also a triple.
Does the Pythagorean theorem work for non-right triangles?
No. For other triangles, use the Law of Cosines: c^2 = a^2 + b^2 - 2ab*cos(C). The Pythagorean theorem is a special case where angle C = 90 degrees and cos(90) = 0.
How do I find the distance between two points using Pythagorean theorem?
The distance formula is d = sqrt((x2-x1)^2 + (y2-y1)^2). This is the Pythagorean theorem applied to a right triangle formed by the horizontal and vertical differences.