Standard Form to General Form of a Circle Calculator

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

Quick Facts

Standard form
(x − h)² + (y − k)² = r²
h, k is the center; r is the radius.
General form
x² + y² + Dx + Ey + F = 0
Expanded, all terms moved to one side.
Conversion formulas
D = −2h, E = −2k, F = h² + k² − r²
Derived by expanding the squared binomials.

Your Results

Calculated
D coefficient
-
D = −2h
E coefficient
-
E = −2k
F constant
-
F = h² + k² − r²
General form equation
-
x² + y² + Dx + Ey + F = 0

Ready

Enter the center (h, k) and radius r, then press Calculate.

Formula and method for Standard Form to General Form of a Circle

A circle's standard form, (x − h)² + (y − k)² = r², is the most direct way to read off the center (h, k) and radius r by inspection. The general form, x² + y² + Dx + Ey + F = 0, is the fully expanded version with every term moved to one side and no parentheses — the form you'll often meet first in a textbook problem or when a circle's equation is given without an obvious center and radius. This calculator converts standard form into general form.

How the calculation works

Start with the standard form (x − h)² + (y − k)² = r² and expand each squared binomial: (x − h)² = x² − 2hx + h², and (y − k)² = y² − 2ky + k². Substituting back gives x² − 2hx + h² + y² − 2ky + k² = r². Moving r² to the left side and collecting terms produces x² + y² + (−2h)x + (−2k)y + (h² + k² − r²) = 0, which matches the general form x² + y² + Dx + Ey + F = 0 with:

  • D = −2h
  • E = −2k
  • F = h² + k² − r²

Enter the center coordinates h and k and the radius r, and the calculator computes D, E, and F and assembles the full general-form equation for you.

Common mistakes

  • Sign errors: because D = −2h and E = −2k, a positive center coordinate produces a negative coefficient. A center at (3, −2) gives D = −6 and E = 4, not D = 6 and E = −2.
  • Radius vs. radius squared: F depends on r² (the radius squared), not r itself. Squaring the radius before subtracting is a common source of errors when doing this by hand.
  • Negative radius: a circle's radius must be a positive number; a "negative radius" has no geometric meaning and cannot be used in the standard-form equation.

Real-world applications

  • Analytic geometry and precalculus courses use this conversion to move between the two standard equation representations of a circle.
  • Completing the square in reverse (general to standard) recovers a circle's center and radius from an equation given in expanded form — common in conic-sections problems.
  • Computer graphics and CAD software sometimes store circle/arc equations in general form for use in systems of equations, and convert to standard form for rendering.
  • Physics and engineering problems involving circular paths or boundaries often start from a described center and radius (standard form) and need the expanded equation for algebraic manipulation.

Frequently Asked Questions

How do you convert a circle's standard form to general form?
Start with standard form (x − h)² + (y − k)² = r², expand both squared binomials, and move every term to one side: x² − 2hx + h² + y² − 2ky + k² − r² = 0. Grouping terms gives the general form x² + y² + Dx + Ey + F = 0, where D = −2h, E = −2k, and F = h² + k² − r².
What do D, E, and F represent in the general form of a circle?
D and E encode the center: the center is at (−D/2, −E/2), which equals (h, k). F encodes both the center and the radius through F = h² + k² − r², so the radius can be recovered as r = √(h² + k² − F).
Can I convert back from general form to standard form?
Yes. From x² + y² + Dx + Ey + F = 0, complete the square on the x-terms and y-terms to get h = −D/2, k = −E/2, and r = √(h² + k² − F). This calculator performs the forward conversion (standard to general); reversing these formulas gives you the standard form.
What if the radius is zero or negative?
A valid circle needs a positive radius (r > 0). A radius of 0 collapses the equation to a single point, and a negative radius is not geometrically meaningful, so the calculator requires r to be greater than zero.

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