General Form of the Equation of a Circle Calculator

Enter a circle's center (h, k) and radius r to convert the standard (center-radius) form into the general form x² + y² + Dx + Ey + F = 0.

Quick Facts

General form
x² + y² + Dx + Ey + F = 0
A circle's equation with the squared binomials expanded and collected into coefficients D, E, and F.
Standard (center-radius) form
(x - h)² + (y - k)² = r²
The form this calculator starts from, using center (h, k) and radius r.
Coefficient relationships
D = -2h, E = -2k, F = h² + k² - r²
Derived by expanding the standard form and matching coefficients term by term.

Your Results

Calculated
General Form Equation
-
x² + y² + Dx + Ey + F = 0
Coefficient D
-
D = -2h
Coefficient E
-
E = -2k
Coefficient F
-
F = h² + k² - r²

Ready

Enter the center coordinates and radius, then press Calculate.

Formula and Method for the General Form of a Circle's Equation

Every circle can be written in standard (center-radius) form, (x - h)² + (y - k)² = r², where (h, k) is the center and r is the radius. Expanding that equation and collecting terms produces the general form, x² + y² + Dx + Ey + F = 0. This calculator takes the center and radius you already know and expands them into the general form automatically, showing the coefficients D, E, and F along with the full equation.

How the calculation works

Starting from (x - h)² + (y - k)² = r², expand both squared binomials: x² - 2hx + h² + y² - 2ky + k² = r². Moving r² to the left side and grouping the linear and constant terms gives x² + y² + (-2h)x + (-2k)y + (h² + k² - r²) = 0. Matching that against x² + y² + Dx + Ey + F = 0 gives three simple formulas: D = -2h, E = -2k, and F = h² + k² - r². The calculator plugs your center coordinates and radius into these three formulas and assembles the final equation string.

Common mistakes

  • Sign errors on D and E: D and E carry the opposite sign of 2h and 2k. A center at (3, -2) gives D = -6 (not +6) and E = +4 (not -4) — double-check the sign before writing the final equation.
  • Using diameter instead of radius: F depends on r², so entering the diameter where the radius is expected inflates F by a factor of four.
  • Forgetting the coefficient of x² and y² must be equal: the general form x² + y² + Dx + Ey + F = 0 assumes both squared terms have a coefficient of 1. If your equation has Ax² + Ay² + ... with A ≠ 1, divide every term by A first.

Real-world applications

  • Analytic geometry and precalculus coursework uses this conversion to move between the intuitive center-radius view and the algebraic general form.
  • Computer graphics and CAD systems often store circles and arcs as general-form conic coefficients for use in intersection and collision formulas.
  • Surveying and machine-path planning sometimes require the general form to feed into standard conic-section solvers.
  • Reversing the process (completing the square on a general-form equation) recovers the center and radius for graphing or measurement.

Frequently Asked Questions

What is the general form of a circle's equation?
The general form is x² + y² + Dx + Ey + F = 0, where D = -2h, E = -2k, and F = h² + k² - r², with (h, k) the center and r the radius. It is the standard (center-radius) form expanded and collected into linear coefficients.
How do I convert standard (center-radius) form to general form?
Start with (x - h)² + (y - k)² = r², expand both squared binomials to get x² - 2hx + h² + y² - 2ky + k² = r², then move r² to the left side. The result is x² + y² + (-2h)x + (-2k)y + (h² + k² - r²) = 0, matching D, E, and F above.
How do I go back from general form to center and radius?
Complete the square in reverse: h = -D/2, k = -E/2, and r = √(h² + k² - F). If h² + k² - F is negative, the equation does not describe a real circle; if it equals zero, the equation represents a single point rather than a circle.

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