How to use the circle equation calculator
This calculator handles the equation of a circle in both standard and general form. In the first section you enter the center and radius to get both forms at once; in the second you enter the general-form coefficients D, E and F to recover the center and radius.
With center (a, b) and radius r the standard form is (x − a)² + (y − b)² = r², and expanding it gives D = −2a, E = −2b and F = a² + b² − r². Terms with a coefficient of 0 or 1 are tidied away so the result reads like x² + y² − 6x + 4y − 12 = 0, and no thousands separators are used inside an equation.
Going the other way, the center is (−D/2, −E/2) and the squared radius is D²/4 + E²/4 − F. When that value is 0 the radius is 0 and only a point remains; when it is negative there is no circle in real coordinates. Both cases stop the calculation and explain what happened.
From the center and radius the calculator also reports the area πr² and the circumference 2πr, and it says whether the origin lies inside, on or outside the circle, printing the distance to the center alongside the radius. Deciding whether a point is exactly on the circle uses a tolerance of 1e-9 rather than exact equality, and values are shown to six decimal places.
Frequently Asked Questions
With center (a, b) and radius r, the standard form is (x − a)² + (y − b)² = r². Expanding it gives the general form x² + y² + Dx + Ey + F = 0 with D = −2a, E = −2b and F = a² + b² − r².
The center is (−D/2, −E/2) and the squared radius is D²/4 + E²/4 − F. A radius exists only when that value is positive.
When D²/4 + E²/4 − F equals 0 the radius is 0, leaving a single point, and when it is negative no circle exists in real coordinates. Both cases show a message instead of a result.