Find the Center and Radius of a Circle from Its Equation

⚪ Circle Equation Calculator: Center & Radius with Step-by-Step Solutions

Convert the general form \(x^2 + y^2 + ax + by = c\) to standard form \((x-h)^2 + (y-k)^2 = r^2\) by completing the square. Complete step-by-step explanation shown for every calculation.
General form: \(x^2 + y^2 + ax + by = c\)
\[ x^2 + y^2 + a x + b y = c \]

Enter coefficients \(a\), \(b\), and \(c\) below.

Results
Enter coefficients and click "Find Center & Radius"
Center \((h, k)\)
( — , — )
Radius \(r\)
—
\((x - h)^2 + (y - k)^2 = r^2\)
📐 Step-by-step solution will appear here after calculation.

Completing the Square Method (Reference)

Given \(x^2 + y^2 + ax + by = c\):

\[ x^2 + ax = \left(x + \frac{a}{2}\right)^2 - \left(\frac{a}{2}\right)^2 \] \[ y^2 + by = \left(y + \frac{b}{2}\right)^2 - \left(\frac{b}{2}\right)^2 \]

Substitute and rearrange:

\[ \left(x + \frac{a}{2}\right)^2 + \left(y + \frac{b}{2}\right)^2 = c + \left(\frac{a}{2}\right)^2 + \left(\frac{b}{2}\right)^2 \]

Thus center \( (h,k) = \left(-\frac{a}{2}, -\frac{b}{2}\right) \) and \( r^2 = c + \left(\frac{a}{2}\right)^2 + \left(\frac{b}{2}\right)^2 \).

More References

  • Equation of a Circle
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