Free · Mathematical relations and geometry

Circle Standard, General and Parametric Equations

Convert a circle center and squared radius between standard, general and parametric coefficient forms.

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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.

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Formula and method

A circle centered at (2, −3) with radius 5 has standard equation (x−2)²+(y+3)²=25 and general equation x²+y²−4x+6y−12=0.

(x−a)²+(y−b)²=r²; D=−2a, E=−2b, F=a²+b²−r²

Worked example

Enter these known values and leave the other values blank.

A
2
B
-3
C
25
A
2
B
-3
r
5
D
-4
E
6
F
-12
x
2
y
-3
Radius
5
Diameter
10
Area
78.5
Circumference
31.4

Assumptions and limitations

  • Standard-form c is radius squared, not the radius. Positive radius describes a nondegenerate real circle.
  • General form has leading coefficients of x² and y² equal to one and no xy term. Normalize a common nonzero leading coefficient before using these controls.
  • Coordinates and radius share a common length unit; area uses its square. The interface does not provide unit selectors.
  • Parametric form uses x=a+r cos α and y=b+r sin α.

Common questions

How do I find the radius from the general form?

Complete the square: r²=(D²+E²)/4−F. The resulting square must be positive.

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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