Free · Circle calculators ⭕

Circle Equations, Center, Radius & Geometry

Connect a circle’s center and radius with standard, general and parametric coefficients, area and circumference.

  • 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

Center (2, −3) and radius 5 give standard form (x − 2)² + (y + 3)² = 25, diameter 10, circumference 10π and area 25π.

C = r²; D = −2A; E = −2B; F = A² + B² − r²; diameter = 2r; circumference = 2πr; area = πr²

Worked example

Enter these known values and leave the other values blank.

Circle center x
2
Circle center y
-3
Known positive radius
5
Standard A: center x
2
Standard B: center y
-3
Standard C: radius squared
25
Parametric A: x offset
2
Parametric B: y offset
-3
Parametric radius
5
General D: coefficient of x
-4
General E: coefficient of y
6
General F: constant
-12
Geometric radius
5
Circle diameter
10
Circle area
78.5
Circle circumference
31.4

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.
  • The parametric coefficients describe x = A + r cos t and y = B + r sin t. The two radius fields refer to the same positive radius, shown in different sections.

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 I start with a diameter or area?

Yes. A positive geometric measurement can determine the radius. The center still requires coordinates or the corresponding equation coefficients.

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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