Center and radius
For center (2, −3) and radius 5:
(x − 2)² + (y + 3)² = 25
x² + y² − 4x + 6y − 12 = 0
Equation of a Circle Calculator From Center, Radius or Three Points
Enter the circle data
Enter finite integers or decimals, including scientific notation. Coordinates may be negative; the radius must be greater than zero.
Circle equation and properties
Standard form
(x − h)² + (y − k)² = r²
General form
x² + y² + Dx + Ey + F = 0
Center—
Radius—
Diameter—
Circumference—
Area—
General coefficients—
Calculation steps and verification
Enter valid values to see the calculation.
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Coordinate graph
The graph rescales automatically. Numerical results remain available if SVG is not displayed.
Equation of a circle formulas
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.
How to find a circle through three points
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.
- Form vectors u = B − A and v = C − A.
- Check uₓvᵧ − uᵧvₓ. A zero value means the points are collinear.
- Solve 2u · c = |u|² and 2v · c = |v|² for the center offset c.
- Add the offset to point A, then calculate the radius from the distance between the center and A.
- Substitute the center and radius into standard form and expand for general form.
Nearly collinear points can imply an extremely large, numerically unstable circle. The tool flags them rather than showing a misleading result.
Worked examples
Center and point
Center (1, 2) and point (4, 6) give r = √((4−1)²+(6−2)²) = 5.
(x − 1)² + (y − 2)² = 25
Three points
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
How to interpret the equation
| 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.
Frequently asked questions
What is the standard equation of a circle?
A circle centered at (h, k) with radius r has equation (x − h)² + (y − k)² = r².
How do I find the equation from a center and radius?
Replace h, k, and r in standard form with the known values. Expand the squared binomials only if you need general form.
How do I find a circle through three points?
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.
Why are collinear points rejected?
No finite circle passes through three distinct points on one straight line. Their perpendicular bisectors cannot identify one finite center.
Can the radius be zero?
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.
Why might a nearly collinear set be rejected?
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.
Does the calculator upload my coordinates?
No. Calculations, graph rendering, clipboard copying, and text download all occur locally in your browser.
Methodology and numerical limits
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.