Standard Form to General Form of a Circle Calculator
Convert (x – h)² + (y – k)² = C into x² + y² + Dx + Ey + F = 0, with every coefficient and expansion step shown.
Standard-form inputs
Required x-coordinate of the center. Use a dot for decimals.
Required y-coordinate of the center; negative values are allowed.
Required nonnegative value. The radius is √C.
General-form result
x² + y² – 6x + 4y – 12 = 0
General equation: x squared plus y squared minus 6 x plus 4 y minus 12 equals zero.
Coefficient derivation
| Coefficient | Rule | Substitution | Result |
|---|---|---|---|
| D | D = – 2h | – 2 × 3 | – 6 |
| E | E = – 2k | – 2 × ( – 2) | 4 |
| F | F = h² + k² – C | 3² + ( – 2)² – 25 | – 12 |
The squared terms keep coefficient 1. Expanding the two binomials creates the linear coefficients D and E, while all constant terms combine into F.
Equivalence check
| Point on circle | Standard-form left side | Target C | General-form left side |
|---|---|---|---|
| (8, – 2) | 25 | 25 | 0 |
| (3, 3) | 25 | 25 | 0 |
| ( – 2, – 2) | 25 | 25 | 0 |
These points are generated from the current center and radius. Each satisfies both forms of the same circle equation, subject only to displayed rounding.
How to use the standard form to general form of a circle calculator
What this calculator does
This calculator expands a circle equation written as (x – h)² + (y – k)² = C and collects like terms to produce x² + y² + Dx + Ey + F = 0. It identifies the center, radius, three general-form coefficients, the expanded intermediate equation, and several points that verify the two equations are equivalent. The result is an exact algebraic conversion for finite real inputs. It does not graph the circle, solve for intersections with another curve, or decide whether a separately entered general quadratic represents a circle.
When to use it
Use the converter when an algebra or analytic-geometry exercise asks you to expand a circle equation; when a system of equations is easier to manipulate in polynomial form; when you need the coefficients D, E, and F for software or a worksheet; or when you want to check hand expansion before submitting a solution. For the geometric meaning of the center and radius in standard form, review the OpenStax explanation of circle equations.
How to calculate
- The calculator opens with a complete demonstration: h = 3, k = – 2, and C = 25. Its workbook is already validated, so Download Excel is available immediately.
- Replace Center h, Center k, and Radius squared C with the values from your standard-form equation. Results update live; no Calculate button is needed.
- Read the General equation first, then inspect Coefficient D, Coefficient E, Coefficient F, and Radius r. The derivation table shows each substitution, while the equivalence table checks generated points in both forms.
- Select Download Excel to export the current typed inputs, outputs, derivation, and verification rows as a genuine XLSX workbook.
- Select Reset to clear the demonstration values and all calculated content. Reset does not restore the sample. Download Excel remains disabled until all three required fields again contain a complete valid circle.
Input guide
Center h is the required real x-coordinate of the circle's center. Enter a plain decimal, a correctly grouped U.S.-style number, or scientific notation; for example, 3, – 1.5, or 1e3. Increasing h moves the center to the right and makes D = – 2h more negative. A common mistake is reading (x + 3)² as h = 3; because standard form uses subtraction, (x + 3)² means h = – 3.
Center k is the required real y-coordinate of the center and accepts the same number formats; the startup value is – 2. Increasing k moves the center upward and changes E = – 2k. As with h, the sign inside the binomial is opposite the center coordinate: (y + 2)² corresponds to k = – 2.
Radius squared C is required and must be nonnegative. It is the entire right-hand side of the standard equation, not the radius itself. For example, C = 25 gives r = 5. Higher C increases the radius and decreases F by the same amount because F = h² + k² – C. C = 0 is accepted as a degenerate point circle; a negative C is rejected because no real radius has a negative square. Decimal commas such as 1,5 are rejected rather than silently interpreted.
Output guide
General equation is the converted identity in the form x² + y² + Dx + Ey + F = 0. Coefficient D equals – 2h, Coefficient E equals – 2k, and Coefficient F equals h² + k² – C. D and E locate the center indirectly; F combines both center coordinates and the radius squared. Any coefficient may be zero or negative without invalidating the circle. Radius r is √C and is always nonnegative for accepted input. The Center, Radius, and classification pills repeat the most useful geometric interpretation.
The Coefficient derivation table lists each coefficient, its rule, the current substitution, and the resulting value. The Equivalence check table lists a generated point, the standard-form left side, the target C, and the general-form left side. For a correct conversion, the standard side equals C and the general side equals 0. These checks are numerical confirmations, while the coefficient formulas establish exact algebraic equivalence.
Worked example
With h = 3, k = – 2, and C = 25, the standard equation is (x – 3)² + (y + 2)² = 25. Expand: x² – 6x + 9 + y² + 4y + 4 = 25. Move 25 to the left and combine constants: 9 + 4 – 25 = – 12. Therefore D = – 6, E = 4, F = – 12, and the first-open result is x² + y² – 6x + 4y – 12 = 0. The radius is √25 = 5. This matches the general circle form discussed in the Mathematics LibreTexts treatment of circles.
Why the coefficient rules work
Expanding (x – h)² gives x² – 2hx + h², and expanding (y – k)² gives y² – 2ky + k². After subtracting C from both sides, the equation becomes x² + y² – 2hx – 2ky + h² + k² – C = 0. Matching coefficients term by term gives D = – 2h, E = – 2k, and F = h² + k² – C. This is a specialized case of the broader general conic equation summarized by OpenStax, where a circle has equal nonzero coefficients on x² and y² and no xy term after normalization.
Common mistakes
Do not enter the radius in the C field unless it already equals its own square. A radius of 7 means C = 49. Keep track of signs inside the binomials, especially when the center coordinate is negative. Finally, move the right-hand constant to the left before combining constants; leaving C on the right produces an expanded equation, but not the requested zero-on-one-side general form.