General to Standard Form of a Circle Calculator
Convert circle equations from general form (x² + y² + Dx + Ey + F = 0) to standard form ((x-h)² + (y-k)² = r²)
Convert General Form to Standard Form
General Form: x² + y² + Dx + Ey + F = 0
Coefficient of x
Coefficient of y
Constant term
x² + y² + + + = 0
Standard Form Result
Standard Form
Invalid Circle
The equation does not represent a valid circle. The radius squared (r²) must be positive. Current r² = 0.00
Step-by-Step Solution
Step 1: Start with the general form
Step 2: Move the constant to the right side
Step 3: Group x and y terms
Step 4: Complete the square for x terms
Take half of D: 0/2 = 0.00
Square it: (0.00)² = 0.00
Step 5: Complete the square for y terms
Take half of E: 0/2 = 0.00
Square it: (0.00)² = 0.00
Step 6: Add completing terms to both sides
Step 7: Factor the perfect square trinomials
Example Calculations
Example 1: Simple Circle
Given: x² + y² - 6x + 4y - 12 = 0
Here: D = -6, E = 4, F = -12
Solution:
h = -(-6)/2 = 3
k = -(4)/2 = -2
r² = 3² + (-2)² - (-12) = 9 + 4 + 12 = 25
Standard form: (x - 3)² + (y + 2)² = 25
Center: (3, -2), Radius: 5
Example 2: Circle at Origin
Given: x² + y² - 25 = 0
Here: D = 0, E = 0, F = -25
Solution:
h = -0/2 = 0
k = -0/2 = 0
r² = 0² + 0² - (-25) = 25
Standard form: (x - 0)² + (y - 0)² = 25 or x² + y² = 25
Center: (0, 0), Radius: 5
Conversion Formulas
From General Form
To Standard Form
Where:
Completing the Square Method
Move constant term to right side
Group x and y terms separately
Take half of linear coefficients
Square the halved coefficients
Add to both sides to complete squares
Factor the perfect square trinomials
Quick Tips
Standard form reveals center and radius directly
For a valid circle, r² must be positive
Complete the square for both x and y terms
Remember to balance the equation when adding terms
Understanding Circle Equation Conversion
General Form
The general form x² + y² + Dx + Ey + F = 0 is the expanded version of a circle equation. While it contains all the necessary information, the center and radius are not immediately apparent.
Why Convert to Standard Form?
- •Center coordinates are clearly visible
- •Radius can be determined immediately
- •Easier to graph and visualize
- •Simplifies geometric calculations
Completing the Square Method
Completing the square is a powerful algebraic technique that transforms a quadratic expression into a perfect square trinomial plus a constant.
Key Formula
x² + bx = (x + b/2)² - (b/2)²
This method allows us to rewrite the general form as a sum of perfect squares, revealing the circle's center and radius in the standard form.