General to Standard Form Circle Calculator
Enter D, E and F from x² + y² + Dx + Ey + F = 0 to find the centre (h, k) and radius r by completing the square — with a full step-by-step walkthrough.
Start: general form
Move the constant F to the right side
Complete the square in x: add (D/2)² = 4
Complete the square in y: add (E/2)² = 9
Standard form — centre (h, k) and radius r
- 1
Centre x (h = −D ÷ 2)
−(-4) ÷ 2 = 2 - 2
Centre y (k = −E ÷ 2)
−(6) ÷ 2 = -3 - 3
r² = D²÷4 + E²÷4 − F
4 + 9 − (-12) = 25This is the discriminant; the circle is real only when r² > 0. - 4
Radius r = √(r²)
√25 = 5
How does this calculator work?
Given x² + y² + Dx + Ey + F = 0: centre is (−D/2, −E/2) and radius is √(D²/4 + E²/4 − F). Complete the square by adding (D/2)² and (E/2)² to both sides. The standard form is (x−h)² + (y−k)² = r². The equation has a real circle only when D²/4 + E²/4 − F > 0.
Formula
How this is calculated
The general form x² + y² + Dx + Ey + F = 0 can be converted to the intuitive standard form (x−h)² + (y−k)² = r² by the technique of completing the square. Move F to the right: x² + Dx + y² + Ey = −F. For the x terms, add (D/2)² to both sides to form a perfect square: (x + D/2)². Do the same for y: (y + E/2)². The right side becomes −F + (D/2)² + (E/2)², which equals r². The centre is (h, k) = (−D/2, −E/2) and the radius is r = √(D²/4 + E²/4 − F).
The step-by-step solution shows each manipulation clearly. If the expression under the square root (D²/4 + E²/4 − F) is negative or zero, the equation does not represent a real circle — it represents a point (r = 0) or has no real locus.
This technique applies any time a conic in general form needs to be identified and graphed. For circles, the two squared terms always have equal coefficients of 1, so completing the square is straightforward. For ellipses (unequal coefficients) or other conics, additional steps are required.
Frequently asked questions
Completing the square rewrites x² + Dx as (x + D/2)² − (D/2)². You add (D/2)² to both sides of the equation so the x terms form a perfect square trinomial, then factor it into a binomial squared.
If the discriminant equals 0, the equation represents a single point (a degenerate circle with radius 0). If it is negative, there are no real solutions — the equation is imaginary and represents no geometric circle in the coordinate plane.
Expand (x−h)² + (y−k)² = r² to get x² − 2hx + h² + y² − 2ky + k² = r², then move everything to the left. D = −2h, E = −2k, F = h² + k² − r². Use the companion general-form-equation-of-a-circle calculator.
Also known as
TG we-Calculate Editorial Team. (2026). General to Standard Form Circle Calculator [Online calculator]. TG we-Calculate. https://we-calculate.com/calculator/general-to-standard-form-circle-calculator
TG we-Calculate Editorial Team. "General to Standard Form Circle Calculator." TG we-Calculate. 2026. https://we-calculate.com/calculator/general-to-standard-form-circle-calculator.
TG we-Calculate Editorial Team, "General to Standard Form Circle Calculator," TG we-Calculate, 2026. [Online]. Available: https://we-calculate.com/calculator/general-to-standard-form-circle-calculator
@misc{wecalculate_general_to_standard_form_circle_calculator, title = {General to Standard Form Circle Calculator}, author = {{TG we-Calculate Editorial Team}}, howpublished = {\url{https://we-calculate.com/calculator/general-to-standard-form-circle-calculator}}, year = {2026}, note = {TG we-Calculate} }
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