General to Standard Form of a Circle Calculator
Enter the coefficients D, E, and F from the general form of a circle equation (x² + y² + Dx + Ey + F = 0) and this tool converts it to standard form (x - h)² + (y - k)² = r². You get the center coordinates, radius, the standard-form equation written out, and the full completing-the-square working so you can follow every step.
Formula
Worked example
For x² + y² + 22x - 12y - 8 = 0: D = 22, E = -12, F = -8. Half-coefficients: 11 and -6. Complete the square: add 11² = 121 for x, add (-6)² = 36 for y. Right side: 8 + 121 + 36 = 165. Standard form: (x + 11)² + (y - 6)² = 165. Center = (-11, 6), radius = sqrt(165) ≈ 12.845.
What is the general form of a circle equation?
The general form of a circle equation is x² + y² + Dx + Ey + F = 0, where D, E, and F are real number coefficients. This form is compact and useful algebraically, but it does not immediately reveal the center or radius of the circle. When you are given an equation in this form, you must transform it into standard form to read off those key geometric properties. The coefficients come up naturally when you expand a standard-form equation or when solving systems of equations involving circles.
What is the standard form and why does it matter?
The standard form of a circle equation is (x - h)² + (y - k)² = r², where (h, k) is the center and r is the radius. This form is called standard because the center and radius are immediately visible without further algebra. It is the starting point for graphing, for checking whether a point lies inside or outside the circle, and for finding tangent lines. Converting from general form to standard form is therefore one of the most common tasks in coordinate geometry and appears in algebra, pre-calculus, and analytic geometry courses.
How to convert: completing the square
The conversion method is called completing the square, applied separately to the x-terms and the y-terms. Start with x² + y² + Dx + Ey + F = 0. Move F to the right: x² + Dx + y² + Ey = -F. For the x-group, compute (D/2)²: this is the term needed to make x² + Dx + (D/2)² a perfect square trinomial equal to (x + D/2)². Add (D/2)² to both sides. Do the same for y using (E/2)². The left side becomes (x + D/2)² + (y + E/2)² and the right side is (D/2)² + (E/2)² - F, which equals r². The center is h = -D/2 and k = -E/2; the radius is r = sqrt(r²). If r² turns out to be zero, the equation represents a single point; if negative, no real circle exists.
Checking whether a valid circle exists
Not every equation of the form x² + y² + Dx + Ey + F = 0 defines a real circle. After completing the square, you get r² = (D/2)² + (E/2)² - F. Three cases arise: if r² is positive, you have a proper circle; if r² is zero, the equation represents a single point (a degenerate circle with radius zero); if r² is negative, there are no real solutions and the equation has no geometric representation in the real plane. This calculator detects all three cases and warns you when the inputs do not define a real circle.
Completing the square: quick reference
| General form term | Completion square to add | Factored result |
|---|---|---|
| x² + Dx | (D/2)² | (x + D/2)² |
| y² + Ey | (E/2)² | (y + E/2)² |
| Center h | -D/2 | x-coordinate of center |
| Center k | -E/2 | y-coordinate of center |
| Radius squared r² | (D/2)² + (E/2)² - F | must be > 0 for a real circle |
The completing the square formulas used to convert general form to standard form.
Frequently asked questions
What is the formula to go from general form to standard form?
Starting from x² + y² + Dx + Ey + F = 0, complete the square on x and y separately. The result is (x + D/2)² + (y + E/2)² = (D/2)² + (E/2)² - F. The center is h = -D/2 and k = -E/2, and the radius is r = sqrt((D/2)² + (E/2)² - F). This calculator does all the arithmetic for you and shows each step.
Why do I need to complete the square?
The general form groups x² and y² together with linear terms in a way that hides the center and radius. Completing the square rewrites each group as a squared binomial, making the center and radius explicit. Once the equation is in standard form (x - h)² + (y - k)² = r², you can read off both values directly without solving any additional equations.
What if r² is negative or zero?
If r² is negative, the equation has no real solution and does not describe any circle on the coordinate plane - this is called an imaginary circle. If r² is exactly zero, the equation represents a single point at (h, k) rather than a proper circle, sometimes called a degenerate or point circle. Both situations indicate that the given values of D, E, and F do not form a valid circle. Adjust your coefficients so that (D/2)² + (E/2)² - F is strictly positive.
What is the difference between the general form and the standard form?
The general form x² + y² + Dx + Ey + F = 0 is a compact algebraic representation that arises when you expand or combine circle equations. The standard form (x - h)² + (y - k)² = r² is the geometric representation: the center (h, k) and radius r are written directly into the equation. Standard form is better for graphing and geometry; general form is often more convenient for algebra and elimination.
Can I reverse this and go from standard form back to general form?
Yes. Expand (x - h)² + (y - k)² = r² by squaring each binomial and rearranging: x² - 2hx + h² + y² - 2ky + k² - r² = 0. The coefficients are D = -2h, E = -2k, and F = h² + k² - r². Use our Standard Form to General Form calculator to do this in the opposite direction.
Does this calculator work for circles not centered at the origin?
Yes. The general form handles any circle in the plane, including those with centers off the origin. When D and E are both zero, the center is at (0, 0) and F = -r², which is the special case of a circle centered at the origin. Any non-zero D or E shifts the center away from the origin.