Equation of a Circle Calculator
Find standard form, general form, and circle properties from various inputs
Calculate Circle Equation
Circle Equation Forms
Standard Form
(x - h)² + (y - k)² = r²
Center: (h, k), Radius: r
General Form
x² + y² + Dx + Ey + F = 0
Expanded form with coefficients
Parametric Form
x = h + r cos(t)
y = k + r sin(t)
Where t is the parameter (angle)
Example Calculations
Example 1: Center & Radius
Given: Center (2, -3), Radius 5
Standard: (x - 2)² + (y + 3)² = 25
General: x² + y² - 4x + 6y - 12 = 0
Example 2: General Form
Given: x² + y² + 6x - 4y + 9 = 0
Center: (-3, 2)
Radius: 2
Circle Tips
Standard form shows center and radius directly
General form is useful for algebraic manipulation
Three non-collinear points uniquely determine a circle
D² + E² - 4F > 0 for a valid circle in general form
Understanding Circle Equations
Standard Form (x - h)² + (y - k)² = r²
The standard form directly shows the circle's center (h, k) and radius r. This form is most intuitive because you can immediately identify the circle's key properties.
Converting Forms
- •Standard to General: Expand the squared terms
- •General to Standard: Complete the square
- •Three points: Use perpendicular bisectors
General Form x² + y² + Dx + Ey + F = 0
The general form is useful for algebraic operations. To verify it represents a circle, check that the discriminant D² + E² - 4F is positive.
Key Relationships
Center: h = -D/2, k = -E/2
Radius: r = √((D² + E² - 4F)/4)
Area: A = πr²
Circumference: C = 2πr