General Form of the Equation of a Circle Calculator
This calculator converts a circle equation between general form and standard form in both directions. Enter the D, E, F coefficients to find the center, radius, and all circle properties; or enter the center coordinates and radius to get the general form coefficients. The steps panel shows every algebraic step with your actual numbers.
Formula
Worked example
Given x² + y² - 4x - 6y + 9 = 0 (D = -4, E = -6, F = 9): h = -(-4)/2 = 2, k = -(-6)/2 = 3, r² = 4 + 9 - 9 = 4, r = 2. Standard form: (x - 2)² + (y - 3)² = 4. Area = π(4) = 12.566, Circumference = 2π(2) = 12.566.
What is the general form of a circle equation?
A circle can be written in two equivalent algebraic forms. The standard form (x - h)^2 + (y - k)^2 = r^2 makes the center (h, k) and radius r immediately visible. The general form x^2 + y^2 + Dx + Ey + F = 0 is the expanded version of standard form: you square out the binomials and rearrange everything to one side. The general form is what you typically encounter when a system of equations produces a second-degree equation in x and y, or when software outputs an equation without simplifying it. Both forms describe the exact same circle; they are just different ways of writing the same relationship.
How to convert general form to standard form (completing the square)
To convert x^2 + y^2 + Dx + Ey + F = 0 to standard form, complete the square separately for the x terms and the y terms. The x terms are x^2 + Dx; add and subtract (D/2)^2 to make a perfect square. The y terms are y^2 + Ey; add and subtract (E/2)^2 similarly. After rearranging, you get (x - (-D/2))^2 + (y - (-E/2))^2 = (D/2)^2 + (E/2)^2 - F. So the center is h = -D/2, k = -E/2, and the radius squared is r^2 = h^2 + k^2 - F. If r^2 is negative, no real circle exists; if r^2 = 0, the "circle" degenerates to a single point.
How to convert standard form to general form
Going the other direction is straightforward algebra. Start with (x - h)^2 + (y - k)^2 = r^2. Expand the squares: x^2 - 2hx + h^2 + y^2 - 2ky + k^2 = r^2. Move r^2 to the left side: x^2 + y^2 - 2hx - 2ky + (h^2 + k^2 - r^2) = 0. Reading off the coefficients gives D = -2h, E = -2k, and F = h^2 + k^2 - r^2. This form is useful in analytic geometry when you need to test whether a point lies on a circle or find the intersection of two circles.
Circle properties and the parametric form
Once you know the center (h, k) and radius r, all other properties follow directly. Diameter = 2r. Area = pi * r^2. Circumference = 2 * pi * r. The parametric form expresses the x and y coordinates as functions of an angle alpha: x = h + r * cos(alpha), y = k + r * sin(alpha), where alpha ranges from 0 to 2*pi. This parametric representation is especially useful in computer graphics, animation, and numerical methods, because it lets you trace any point on the circle by varying alpha.
Key circle equation formulas
| Property | General form formula | Example (D=-4, E=-6, F=9) |
|---|---|---|
| Center x (h) | h = -D / 2 | h = -(-4)/2 = 2 |
| Center y (k) | k = -E / 2 | k = -(-6)/2 = 3 |
| Radius squared | r² = h² + k² - F | r² = 4 + 9 - 9 = 4 |
| Radius | r = sqrt(h² + k² - F) | r = sqrt(4) = 2 |
| D from center | D = -2h | D = -2(2) = -4 |
| E from center | E = -2k | E = -2(3) = -6 |
| F from center/radius | F = h² + k² - r² | F = 4 + 9 - 4 = 9 |
| Area | πr² | π(4) = 12.566 sq units |
| Circumference | 2πr | 2π(2) = 12.566 units |
| Diameter | 2r | 2(2) = 4 units |
Relationships between general form coefficients (D, E, F) and standard form parameters (h, k, r).
Frequently asked questions
What does it mean if r^2 is negative in the general form?
If the value h^2 + k^2 - F computes to a negative number, the equation has no real solution and no real circle exists. This is sometimes called an imaginary circle. It means the original equation x^2 + y^2 + Dx + Ey + F = 0 is not satisfied by any real (x, y) pair. Check your D, E, and F values for errors.
What happens when r^2 = 0?
When h^2 + k^2 - F = 0 exactly, the equation represents a single point (h, k) rather than a circle. This is called a degenerate or point circle. The radius is zero, so the circumference and area are both zero.
How do I find the radius from the general form without completing the square?
Use the shortcut formula: r = sqrt((D/2)^2 + (E/2)^2 - F). This comes directly from completing the square but skips the intermediate algebra. For example, with D = -4, E = -6, F = 9: r = sqrt(4 + 9 - 9) = sqrt(4) = 2.
Can any circle be written in general form?
Yes, every circle in the Cartesian plane has a general form equation x^2 + y^2 + Dx + Ey + F = 0 with specific real values of D, E, and F. A circle centered at the origin with radius r has D = 0 and E = 0, simplifying to x^2 + y^2 + F = 0 where F = -r^2.
What is the parametric form of a circle and when is it useful?
The parametric form writes x and y as functions of a single angle: x = h + r*cos(alpha), y = k + r*sin(alpha). As alpha sweeps from 0 to 2*pi, the point (x, y) traces out the full circle exactly once. This representation is essential in computer graphics for drawing circles, in robotics for path planning, and in calculus for computing arc length and surface area of revolution.
How is the general form different from the standard form?
Standard form (x - h)^2 + (y - k)^2 = r^2 makes the center and radius explicit. General form x^2 + y^2 + Dx + Ey + F = 0 hides them inside the coefficients. Both describe the same circle; you convert between them by completing the square (general to standard) or expanding and rearranging (standard to general).