Free · Circle calculators ⭕

Circle General ↔ Standard Form Coefficients

Convert the coefficients of a normalized circle equation to or from its center and squared radius.

  • Formula & worked example
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Calculator inputs

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How to use this calculator

  1. Enter the known values in the units shown. Results update as you type.
  2. Where results are editable, change one to solve backwards. Lock a value to hold it fixed.
  3. Use the worked example to check the method. Reset restores the starting fields.

Private by default

Inputs and results stay in this browser tab. Bookify does not upload or store the values you enter.

Formula and method

x² + y² − 4x + 6y − 12 = 0 becomes (x − 2)² + (y + 3)² = 25. Its center is (2, −3) and its radius is 5.

D = −2A; E = −2B; F = A² + B² − C; (x − A)² + (y − B)² = C

Worked example

Enter these known values and leave the other values blank.

General form D: coefficient of x
-4
General form E: coefficient of y
6
General form F: constant
-12
Standard form A: center x
2
Standard form B: center y
-3
Standard form C: radius squared
25

Assumptions and limitations

  • The standard equation is (x − A)² + (y − B)² = C, where C is radius squared. The center is (A, B); C is not the radius.
  • The normalized general equation is x² + y² + Dx + Ey + F = 0. Both squared terms must have coefficient 1 and there must be no xy term. Divide a common nonzero squared-term coefficient out first when applicable.
  • A real nondegenerate circle requires C = (D² + E²)/4 − F > 0. Zero would be a single point, and a negative value gives no real circle; both are excluded.
  • Coordinates use one consistent length scale. Squared radius and area use the square of that scale. A change of coordinate units must be applied consistently to every coefficient.

Common questions

Why is C different from the radius?

The equation equates a sum of squared distances to C. Therefore C = r² and the positive radius is √C; entering r where r² belongs describes a different circle.

Can this normalize an ellipse?

No. Unequal x² and y² coefficients or a cross term require a different conic model. The formulas here assume a circle in normalized form.

References

The calculation equations, inverse formulas, units, and input rules were imported from this source. Bookify provides the interface and equation solver.

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