General Form of the Equation of a Circle Calculator
Convert between general form (x² + y² + Dx + Ey + F = 0) and standard form ((x-A)² + (y-B)² = C)
Circle Equation Converter
General Form: x² + y² + Dx + Ey + F = 0
Coefficient of x
Coefficient of y
Constant term
Equation Forms
General Form
Standard Form
Parametric Form
where α is the parameter (angle in radians)
Invalid Circle
The equation does not represent a valid circle. The radius squared must be positive.
Calculation Steps
Step 1: Calculate center coordinates
A = -D/2 = -0/2 = 0.00
B = -E/2 = -0/2 = 0.00
Step 2: Calculate radius squared
C = A² + B² - F = 0.00² + 0.00² - 0 = 0.00
Example Calculations
Example 1: General to Standard
Given: x² + y² - 6x + 4y - 12 = 0
Here: D = -6, E = 4, F = -12
Solution:
A = -(-6)/2 = 3
B = -(4)/2 = -2
C = 3² + (-2)² - (-12) = 9 + 4 + 12 = 25
Standard form: (x - 3)² + (y + 2)² = 25
Center: (3, -2), Radius: 5
Example 2: Standard to General
Given: (x - 6)² + (y - 6)² = 49
Here: A = 6, B = 6, C = 49
Solution:
D = -2(6) = -12
E = -2(6) = -12
F = 6² + 6² - 49 = 36 + 36 - 49 = 23
General form: x² + y² - 12x - 12y + 23 = 0
Circle Equation Forms
General Form
Expanded form with coefficients
Standard Form
Center-radius form
Parametric Form
y = B + r sin(α)
Parameter-based form
Conversion Formulas
General → Standard
Standard → General
Quick Tips
General form shows all terms expanded
Standard form clearly shows center and radius
For a valid circle, C must be positive
Parametric form is useful for plotting
Understanding Circle Equations
General Form Equation
The general form of a circle equation is x² + y² + Dx + Ey + F = 0, where D, E, and F are constants. This form shows all terms expanded and is useful for algebraic manipulations.
Key Properties
- •Coefficients of x² and y² are always 1
- •No xy term (coefficient of xy is 0)
- •D and E determine the center location
- •F affects the radius of the circle
Standard Form Benefits
The standard form (x - A)² + (y - B)² = C directly shows the center at (A, B) and radius as √C. This form is most intuitive for understanding circle geometry.
Applications
- •Graphing circles on coordinate planes
- •Finding intersections with lines or other circles
- •Computer graphics and geometric modeling
- •Engineering and architectural design