Standard Equation of a Circle Calculator
Find circle equations in standard, general, and parametric forms
Calculate Circle Equation
Example Calculation
Given: Center (3, -2), Radius = 5
Standard Form: (x - 3)² + (y + 2)² = 25
General Form: x² + y² - 6x + 4y - 12 = 0
Parametric Form: x = 3 + 5cos(α), y = -2 + 5sin(α)
Area: π × 5² = 78.54 square units
Circumference: 2π × 5 = 31.42 units
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 standard form
Parametric Form
x = h + r·cos(α)
y = k + r·sin(α)
α is the parameter angle
Conversion Formulas
General to Standard:
h = -D/2
k = -E/2
r = √((D² + E² - 4F)/4)
Standard to General:
D = -2h
E = -2k
F = h² + k² - r²
Quick Tips
Standard form directly shows center and radius
General form coefficients relate to center position
Parametric form is useful for plotting points
Radius must be positive for a valid circle
Understanding Circle Equations
Standard Form
The standard form (x - h)² + (y - k)² = r² immediately reveals the circle's center (h, k) and radius r. This is the most intuitive form for understanding a circle's geometry.
General Form
The general form x² + y² + Dx + Ey + F = 0 is the expanded version of the standard form. It's useful for algebraic manipulations and can represent degenerate cases.
Parametric Form
The parametric form uses an angle parameter α to describe all points on the circle. It's particularly useful for plotting circles and in calculus applications.
Conversion Between Forms
Standard to General
Expand (x-h)² + (y-k)² = r² and collect terms
General to Standard
Complete the square for x and y terms
To Parametric
Use trigonometric identities with center and radius
Circle Properties
Key Measurements:
- • Area = πr²
- • Circumference = 2πr
- • Diameter = 2r
Point Relations:
- • Distance from center = r (on circle)
- • Distance from center < r (inside circle)
- • Distance from center > r (outside circle)