How this calculation works
A circle has equal squared distance from its center to all three points. Subtracting the first distance equation from the other two yields a linear 2 × 2 system for the center. The squared radius is then the squared distance to any entered point.
(x−h)² + (y−k)² = r²A worked example
The points (1,0), (0,1), (−1,0) lie on the circle x² + y² = 1, centered at (0,0).
Good to know
Flat Cartesian coordinates with matching units. Each component is between −1,000,000 and 1,000,000; decimal inputs allow up to 6 places and integer fractions have denominators up to 1,000,000. Expressions are not parsed. Exact intermediate fractions have a size limit; reduce unusually complex inputs if that limit is reached. Lengths and angles are decimal approximations.
Reference: MathWorld: circle equations (opens in a new tab) ↗
Equal distances identify the center
A circle center must have the same squared distance to A, B and C. Subtracting two squared-distance equations cancels the common squared-center terms, leaving a linear relation. Two independent relations determine the center.
Three distinct non-collinear points define one finite circle. Repeated points or collinear points do not provide that information. A large radius is not a substitute for a unique finite circle through three collinear points.
The squared radius remains an exact rational value for supported inputs even when the radius itself is irrational. For example, a squared radius of 2 is exact, while the displayed radius is an approximation of √2.
Nearly collinear measurements can give surprising circles
As the three points approach a line, the inferred center can move far away and the circle can become very large. A small perturbation of measured coordinates may then change the center substantially. Exact arithmetic preserves the entered numbers without resolving that sensitivity.
Check the result by substituting each point into (x−h)² + (y−k)². All three should produce the same r². This is a more useful verification than judging whether a large circle would look plausible on a small screen.
The circle center is the circumcenter of the triangle made from the points. It may lie outside that triangle, particularly for an obtuse triangle. The ordinary vertex-average centroid answers a different question and generally has different coordinates.
A few questions, answered
Can the points be entered clockwise?
Yes. Reversing their order changes intermediate determinant signs but leaves the center and radius unchanged.
Why is there no automatic fit to many noisy points?
This tool uses exactly three points. A best-fit circle for a larger dataset requires a separate fitting method and an error model.
Can I type square roots or variables?
No. Enter rational components as integers, supported finite decimals or fractions. Symbolic coordinate expressions are not parsed.