Circle Equation Formulas

Conversion Between Forms

To convert from standard to general form:

• Expand the standard form: (x – h)² + (y – k)² = r²
• This becomes: x² – 2hx + h² + y² – 2ky + k² = r²
• Rearrange: x² + y² – 2hx – 2ky + (h² + k² – r²) = 0
• So D = -2h, E = -2k, F = h² + k² – r²

To convert from general to standard form:

• Complete the square for x and y terms
• Group x and y terms: (x² + Dx) + (y² + Ey) = -F
• Complete the square: (x² + Dx + (D/2)²) + (y² + Ey + (E/2)²) = -F + (D/2)² + (E/2)²
• This becomes: (x + D/2)² + (y + E/2)² = (D² + E² – 4F)/4
• So center is (-D/2, -E/2) and radius is √[(D² + E² – 4F)/4]