General Form Equation of a Circle Calculator

Formula

How this is calculated

Start with the standard (centre-radius) form of a circle: (x − h)² + (y − k)² = r². Expanding the squared binomials gives x² − 2hx + h² + y² − 2ky + k² = r². Collecting all terms onto the left side produces the general form x² + y² + Dx + Ey + F = 0, where D = −2h, E = −2k and F = h² + k² − r².

The general form is useful when working with systems of equations involving multiple conics, or when fitting a circle to data points using a linear least-squares method — because D, E and F appear linearly in the equation. Converting back to standard form (to recover centre and radius) requires completing the square for x and y, which the companion general-to-standard-form calculator handles.

Note: the equation x² + y² (both squared terms with coefficient 1) represents a circle specifically, not an ellipse. F can be positive, negative or zero depending on the values of h, k and r. The condition for a real circle is that the discriminant D²/4 + E²/4 − F = r² is strictly positive, which is always satisfied here since r > 0.