General to Standard Form of a Circle Calculator

Enter the D, E, and F coefficients from a circle's general-form equation x² + y² + Dx + Ey + F = 0 to get its standard form (x − h)² + (y − k)² = r², center, and radius.

Quick Facts

General form
x² + y² + Dx + Ey + F = 0
D, E, and F are real coefficients; there is no xy term.
Standard form
(x − h)² + (y − k)² = r²
Center (h, k), radius r, found by completing the square.
Center & radius
h = −D/2, k = −E/2, r = √(h² + k² − F)
A negative value under the root means no real circle exists.

Your Results

Calculated
Standard Form Equation
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(x − h)² + (y − k)² = r²
Center (h, k)
-
h = −D/2, k = −E/2
Radius r
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r = √(h² + k² − F)
r² (h² + k² − F)
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Determines whether a real circle exists

Ready

Enter the D, E, and F coefficients, then press Calculate.

Formula and Method for Converting General to Standard Form of a Circle

Every circle can be written two ways: general form, x² + y² + Dx + Ey + F = 0, and standard form, (x − h)² + (y − k)² = r², where (h, k) is the center and r is the radius. General form comes from expanding standard form, so converting back is a matter of undoing that expansion by completing the square on the x-terms and the y-terms separately.

How the calculation works

Start with x² + y² + Dx + Ey + F = 0 and group by variable: (x² + Dx) + (y² + Ey) = −F. To complete the square on x² + Dx, add and subtract (D/2)²: this turns it into (x + D/2)² − (D/2)². Do the same for y² + Ey using (E/2)². Substituting back and moving constants to the right side gives (x + D/2)² + (y + E/2)² = (D/2)² + (E/2)² − F. Comparing this to (x − h)² + (y − k)² = r² shows h = −D/2, k = −E/2, and r² = (D/2)² + (E/2)² − F, so r = √((D/2)² + (E/2)² − F). This calculator applies those three formulas directly: enter D, E, and F, and it reports the center, radius, and the fully written standard-form equation.

Common mistakes

  • Sign errors on the center: the center is (−D/2, −E/2), not (D/2, E/2) — the sign flips when you move terms from inside the completed square to the center coordinates.
  • Forgetting the F constant: the radius depends on F as well as D and E; dropping F from the calculation gives the wrong radius even if the center is correct.
  • Assuming every general-form equation is a circle: if (D/2)² + (E/2)² − F is negative, there is no real radius and the equation has no real graph; if it equals zero, the "circle" is a single point.

Real-world applications

  • Analytic geometry and precalculus courses use this conversion to identify a circle's center and radius from an expanded equation.
  • Computer graphics and CAD software store circles in general (implicit) form for intersection tests, then convert to standard form to render or manipulate them.
  • Physics and engineering problems involving circular motion or boundaries often start from an expanded equation that needs to be converted before the center and radius can be read off.
  • GPS and mapping calculations that define circular coverage areas from quadratic equations rely on the same center/radius extraction.

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, E, and F are real number coefficients. Because the coefficients of x² and y² are equal (both 1) and there is no xy term, this equation always describes a circle (or a point, or no real graph, depending on D, E, and F).
How do you convert general form to standard form?
Group the x-terms and y-terms, then complete the square on each: x² + Dx becomes (x + D/2)² − (D/2)², and y² + Ey becomes (y + E/2)² − (E/2)². Moving the constants to the right side gives the standard form (x − h)² + (y − k)² = r², with center h = −D/2, k = −E/2, and radius r = √(h² + k² − F).
What if h² + k² − F is zero or negative?
If h² + k² − F equals zero, the equation represents a single point (x, y) = (h, k), a circle of radius 0. If h² + k² − F is negative, there is no real radius, so the equation has no real graph — no circle exists for those coefficients.
How do I find the center and radius directly from general form?
You do not need to fully expand the standard form to get the center and radius: the center is always (−D/2, −E/2), and the radius is √((D/2)² + (E/2)² − F). These two formulas come directly from completing the square.

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