Convert a circle’s general-form equation (x² + y² + Dx + Ey + F = 0) into center-radius form.
How it works
The center is at (-D/2, -E/2), and the radius is √((D/2)² + (E/2)² – F) — found by completing the square. D = -4, E = -6, F = 4 gives a circle centered at (2, 3) with radius 3.
What this does not include
This converts general form to center-radius form only. It doesn’t go the reverse direction (center-radius to general form), which would need a separate expansion of the standard equation.
How to use this calculator
- Enter the D, E, and F coefficients from the general-form equation.
A worked example
The general form equation x² + y² − 4x − 6y + 4 = 0 (d=−4, e=−6, f=4) converts to center (2, 3), radius 3.
x² + y² − 25 = 0 (d=0, e=0, f=−25): center (0, 0), radius 5 — a circle centered at the origin.
What the variables mean
| Variable | Meaning |
|---|---|
| d, e, f | Coefficients from the general form x² + y² + dx + ey + f = 0 |
Edge cases worth knowing
This converts general form to the more intuitive center-radius form by completing the square on both x and y terms — the same technique used to convert a general quadratic to vertex form.
Not every d, e, f combination describes a real circle. If the completed-square radius² works out negative (like d=0, e=0, f=25 above), there’s no real circle satisfying the equation, so the calculator declines to show a result.
What does it mean if the calculator returns no result?
The coefficients don’t correspond to a real circle — the radius formula would require taking the square root of a negative number, which happens when the equation actually describes an empty set or a single point.
Why is general form useful if center-radius form is easier to read?
General form results naturally from algebraic manipulation, like finding a circle through three points or intersecting it with another curve, so converting to center-radius form afterward makes the geometry easier to interpret.
What is “completing the square” doing here?
It rewrites the x and y terms as perfect squares, which is exactly the standard center-radius form (x – h)² + (y – k)² = r² — this calculator does that algebra for you.
