Enter finite integers or decimals, including scientific notation. Coordinates may be negative; the radius must be greater than zero.
Enter valid values to see the calculation.
The graph rescales automatically. Numerical results remain available if SVG is not displayed.
A circle is the set of points that are the same distance from a fixed center. If the center is (h, k) and the radius is r > 0, the standard form is:
(x − h)² + (y − k)² = r²
Expanding both squares gives the general form:
x² + y² − 2hx − 2ky + (h² + k² − r²) = 0
Therefore D = −2h, E = −2k, and F = h² + k² − r² in x² + y² + Dx + Ey + F = 0.
Three distinct, non-collinear points define one circle. The calculator translates the first point to the origin and solves two perpendicular-bisector equations. This scaled form reduces overflow and improves stability for very large or very small coordinates.
Nearly collinear points can imply an extremely large, numerically unstable circle. The tool flags them rather than showing a misleading result.
For center (2, −3) and radius 5:
(x − 2)² + (y + 3)² = 25
x² + y² − 4x + 6y − 12 = 0
Center (1, 2) and point (4, 6) give r = √((4−1)²+(6−2)²) = 5.
(x − 1)² + (y − 2)² = 25
The points (0, 0), (6, 0), and (0, 8) form a right triangle. Its circumcenter is the midpoint of the hypotenuse, (3, 4), and r = 5.
(x − 3)² + (y − 4)² = 25
| Result | Meaning | Use |
|---|---|---|
| Standard form | (x − h)² + (y − k)² = r² | Read the center and radius directly. |
| General form | x² + y² + Dx + Ey + F = 0 | Compare coefficients or combine with other equations. |
| Center | (−D/2, −E/2) | Locate the circle on a coordinate plane. |
| Radius | √((D² + E²)/4 − F) | Recover the radius from general form. |
Signs in standard form can look reversed: a center at (2, −3) produces (x − 2)² + (y + 3)².
A circle centered at (h, k) with radius r has equation (x − h)² + (y − k)² = r².
Replace h, k, and r in standard form with the known values. Expand the squared binomials only if you need general form.
Find the intersection of two perpendicular bisectors to get the center, then measure from that center to any supplied point for the radius. The calculator performs the equivalent linear-system calculation.
No finite circle passes through three distinct points on one straight line. Their perpendicular bisectors cannot identify one finite center.
A zero radius describes a single point, usually called a degenerate circle. This calculator requires a positive radius because it is designed for ordinary circles.
Its implied center and radius may be extremely far away, so ordinary floating-point arithmetic can lose meaningful precision. Rejecting numerically unstable input is safer than displaying unreliable coefficients.
No. Calculations, graph rendering, clipboard copying, and text download all occur locally in your browser.
Calculation note: Last reviewed July 30, 2026 by Starlight Tools. Formulas use standard Euclidean analytic geometry. The three-point solver translates and scales coordinates before solving the circumcenter system.
Inputs must be finite numbers with absolute value no greater than 10¹⁰⁰. Display rounding does not change the underlying calculation. Because JavaScript numbers use IEEE 754 double precision, results involving very large, very small, or nearly collinear coordinates should be treated as numerical approximations.
