Standard to General Form Circle Calculator
Switch between standard form (x − h)² + (y − k)² = r² and general form x² + y² + Dx + Ey + F = 0 for any circle. Enter the parameters in either form and get the other form instantly, together with the centre, radius, and an animated diagram.
Conversion direction
- 1
D = −2h
−2 × 2 = -4 - 2
E = −2k
−2 × 3 = -6 - 3
F = h² + k² − r²
2² + 3² − 5² = -12 - 4
Radius squared r²
5² = 25
How does this calculator work?
Standard form (x−h)²+(y−k)²=r² expands to general form x²+y²+Dx+Ey+F=0 using D=−2h, E=−2k, F=h²+k²−r². Reverse it by completing the square: h=−D/2, k=−E/2, r=√(D²/4+E²/4−F). The calculator handles both directions instantly.
Formula
How this is calculated
Standard form makes the geometry explicit: you read the centre (h, k) and radius r directly from the equation. Expanding the squared binomials gives x² − 2hx + h² + y² − 2ky + k² = r², and moving everything to one side produces the general form x² + y² + Dx + Ey + F = 0, where D = −2h, E = −2k, and F = h² + k² − r².
Converting back from general to standard form requires completing the square independently for x and y. Group x² + Dx and add and subtract (D/2)² to form a perfect square (x + D/2)², and do the same for y. The centre emerges as (−D/2, −E/2) and the radius as √(D²/4 + E²/4 − F). If D²/4 + E²/4 − F ≤ 0 there is no real circle — the coefficients describe either a point (= 0) or the empty set (< 0).
Both forms encode the same set of points; standard form is better for geometry (centre, radius visible at a glance), while general form arises naturally when you expand products of polynomials or intersect curves algebraically.
Frequently asked questions
A real circle requires D²/4 + E²/4 − F > 0. If this expression equals zero the equation describes a single point; if it is negative the equation has no real solutions (the circle is imaginary). This calculator warns you and returns no result in those cases.
Completing the square rewrites x² + Dx as (x + D/2)² − (D/2)², turning a quadratic expression into a perfect square plus a constant. Applied to both x and y, it converts the general form into the standard form, making the centre and radius explicit.
Yes — every real circle has a unique general form when the coefficients of x² and y² are both 1 (the normalised form used here). Standard form is also unique once centre and radius are fixed. The two forms are algebraically equivalent and describe identical point sets.
Also known as
TG we-Calculate Editorial Team. (2026). Standard to General Form Circle Calculator [Online calculator]. TG we-Calculate. https://we-calculate.com/calculator/standard-to-general-form-circle-calculator
TG we-Calculate Editorial Team. "Standard to General Form Circle Calculator." TG we-Calculate. 2026. https://we-calculate.com/calculator/standard-to-general-form-circle-calculator.
TG we-Calculate Editorial Team, "Standard to General Form Circle Calculator," TG we-Calculate, 2026. [Online]. Available: https://we-calculate.com/calculator/standard-to-general-form-circle-calculator
@misc{wecalculate_standard_to_general_form_circle_calculator, title = {Standard to General Form Circle Calculator}, author = {{TG we-Calculate Editorial Team}}, howpublished = {\url{https://we-calculate.com/calculator/standard-to-general-form-circle-calculator}}, year = {2026}, note = {TG we-Calculate} }
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