Circle equation calculator guide
A circle is the set of points a fixed distance from a center. In standard form, that distance relationship appears directly as (x - h)^2 + (y - k)^2 = r^2.
The calculator also handles x^2 + y^2 + Dx + Ey + F = 0. Completing both squares gives h = -D/2, k = -E/2, and r^2 = h^2 + k^2 - F. A positive r^2 is required for a nondegenerate real circle.
How to use the circle equation calculator
- Choose the direction: Start with either center and radius or the three general-form coefficients.
- Enter finite values: Use a strictly positive radius. For general form, enter D, E, and F after the squared terms have coefficient 1.
- Calculate: Select Calculate circle to produce both equation forms and the geometric properties.
- Check the graph: Use the labeled diagram as a structural check; it is normalized and not a coordinate-scale plotting tool.
Formula and variables
Expanding the standard form produces the general coefficients. Reversing those relationships is equivalent to completing the square.
(x - h)^2 + (y - k)^2 = r^2; D = -2h; E = -2k; F = h^2 + k^2 - r^2- h — Horizontal center
- The x-coordinate of the center.
- k — Vertical center
- The y-coordinate of the center.
- r — Radius
- Positive distance from the center to every point on the circle.
- D, E, F — General coefficients
- Coefficients in x^2 + y^2 + Dx + Ey + F = 0.
Convert a general equation
Find the center and radius of x^2 + y^2 - 4x - 6y + 4 = 0.
- D
- -4
- E
- -6
- F
- 4
- h = -D/2 = 2
- k = -E/2 = 3
- r^2 = 2^2 + 3^2 - 4 = 9
Result: (x - 2)^2 + (y - 3)^2 = 9, with center (2, 3) and radius 3.
Completing the squares reveals the translated center and the squared radius.
Understanding your results
Standard equation
Read the center with signs reversed inside the binomials. The right side is r squared, not r.
General equation
This expanded form is convenient for coefficient comparisons but does not display the center and radius directly.
Assumptions
- The x-squared and y-squared coefficients are both 1 and there is no xy term.
- All coefficients and geometric inputs are real finite numbers.
- The intended object is a nondegenerate real circle with radius greater than zero.
Limitations
- The calculator does not handle ellipses, spheres, imaginary circles, or equations with unequal squared coefficients.
- The diagram is normalized to show structure rather than exact coordinate-axis scale.
- It does not solve for intersections with lines or other circles.
- Rounded display text can hide very small floating-point residuals although calculations retain full numeric precision.
Common mistakes
- Reading h directly from x - h without reversing a displayed plus sign.
- Using r instead of r squared on the right side.
- Forgetting to divide D and E by negative 2.
- Entering an equation whose x-squared and y-squared coefficients have not been normalized to 1.
- Treating r squared equal to zero as an ordinary circle.
Practical use cases
Analytic geometry
Convert a circle equation into properties needed for graphing or comparison.
Equation construction
Write both equation forms from a specified center and radius.
Planning and decision guide
Normalize first
If both squared terms share a nonzero coefficient other than 1, divide the entire equation by that coefficient before entering D, E, and F. If the coefficients differ or an xy term appears, the conic is not represented by this calculator.
Frequently asked questions
What is the standard equation of a circle?
It is (x - h)^2 + (y - k)^2 = r^2 for center (h, k) and radius r.
How do I find the center from general form?
For x^2 + y^2 + Dx + Ey + F = 0, the center is (-D/2, -E/2).
Why are my coefficients rejected?
They produce r squared less than or equal to zero, so they do not describe a nondegenerate real circle.
Can the center coordinates be negative?
Yes. The center may be anywhere in the coordinate plane; only the radius must be positive.
Does the graph use an exact coordinate scale?
No. It is a normalized explanatory diagram that labels the computed center and radius.
Sources and review
- Distance and Midpoint Formulas; Circles — OpenStax Intermediate Algebra 2e. Accessed 2026-08-29.
- The Ellipse: Circle Review — OpenStax College Algebra with Corequisite Support 2e. Accessed 2026-08-29.
Reviewed 2026-08-29.