General to Standard Form Circle Calculator

Convert a circle equation from general form x² + y² + Dx + Ey + F = 0 to standard form (x - h)² + (y - k)² = r² by completing the square.

Enter General Form Coefficients

Enter D, E, F for x² + y² + Dx + Ey + F = 0

Conversion Result

x² + y² - 4x + 6y - 3 = 0
↓
(x - 2)² + (y + 3)² = 16
Center (h, k) (2, -3)
Radius r 4
r² 16
Diameter 8
Area 50.265482
Circumference 25.132741

Completing the Square - Step by Step

(x - 2)² + (y + 3)² = 16

Converting General Form to Standard Form

The general form of a circle equation is Ax² + Ay² + Dx + Ey + F = 0, where A is the leading coefficient (must be equal for x² and y²). The standard form, (x - h)² + (y - k)² = r², directly reveals the center (h, k) and radius r. Converting between these forms requires completing the square for both the x and y terms.

The Completing the Square Process

Step 1: Normalize

If A is not 1, divide the entire equation by A so the coefficients of x² and y² become 1.

x² + y² + (D/A)x + (E/A)y + (F/A) = 0

Step 2: Group & Rearrange

Move the constant to the right and group x and y terms together.

(x² + Dx) + (y² + Ey) = -F

Step 3: Complete x Square

Take half of D, square it, and add to both sides.

(x + D/2)² = x² + Dx + (D/2)²

Step 4: Complete y Square

Take half of E, square it, and add to both sides.

(y + E/2)² = y² + Ey + (E/2)²

When Does the Equation Not Represent a Circle?

After completing the square, if the right side (r²) is negative, then the equation has no real graph. If r² equals zero, the graph is a single point (a degenerate circle). Only when r² is positive does the equation represent a real circle with a measurable radius.

Special Cases

  • r² > 0: A real circle with center (h, k) and radius r = sqrt(r²).
  • r² = 0: A point circle consisting only of the point (h, k).
  • r² < 0: No real graph exists; the equation has no solution in real numbers.
  • Different coefficients on x² and y²: The equation represents an ellipse, not a circle.

Examples of General to Standard Form

Example 1: x² + y² - 6x + 8y - 11 = 0

Here D = -6, E = 8, F = -11. The center is (3, -4) and r² = 9 + 16 + 11 = 36, so r = 6. Standard form: (x - 3)² + (y + 4)² = 36.

Example 2: 2x² + 2y² + 8x - 12y + 2 = 0

First divide by 2: x² + y² + 4x - 6y + 1 = 0. Then D = 4, E = -6, F = 1. Center is (-2, 3) and r² = 4 + 9 - 1 = 12, so r = 2sqrt(3). Standard form: (x + 2)² + (y - 3)² = 12.

Why This Conversion Matters

  • Standard form is needed for graphing circles on coordinate planes.
  • Finding intersection points between circles and lines requires knowing center and radius.
  • Many geometry proofs rely on the standard form representation.
  • Computer graphics and CAD applications use standard form for circle rendering.