Circle Equation Calculator
Enter D, E, and F—or enter a center and radius—to convert both circle equation forms.
Circle equation forms
The standard form (x − h)² + (y − k)² = r² shows the center (h, k) and radius r directly. The general form used here is x² + y² + Dx + Ey + F = 0. Both squared terms must have coefficient 1 and there is no xy term.
General form to center and radius
For x² + y² − 4x + 6y − 12 = 0, D = −4, E = 6, and F = −12. Then h = −D/2 = 2 and k = −E/2 = −3. Finally, r² = h² + k² − F = 4 + 9 + 12 = 25, so the radius is 5.
Center and radius to general form
Expand the standard form or use D = −2h, E = −2k, and F = h² + k² − r². A center of (2, −3) with radius 5 therefore gives D = −4, E = 6, and F = −12.
Scope and precision
This calculator handles circles aligned to the x-y axes in the normalized forms shown above. It does not parse arbitrary equations, unequal x² and y² coefficients, rotated conics, or circles fitted through three points. Results display up to ten decimal places; preserve the precision appropriate to the original values.
Frequently asked questions
What is the standard equation of a circle?
A circle with center (h, k) and radius r has standard form (x − h)² + (y − k)² = r².
How do I find the center from general form?
For x² + y² + Dx + Ey + F = 0, the center is (−D/2, −E/2). Be careful to reverse the signs of D and E before dividing by two.
How do I find the radius from general form?
Calculate r² = h² + k² − F after finding h and k. The radius is the nonnegative square root of r² when r² is positive.
What if the calculated radius squared is zero?
The equation represents only the center point, so it is a degenerate circle rather than a circle with positive radius.
What if radius squared is negative?
There is no real circle because no real number has a negative square. The equation may still describe a circle over complex coordinates, which is outside this calculator's scope.