Go from centre & radius to the standard equation, or from the general equation back to the centre & radius — pick your direction below.
Circle Equation Calculator
How It’s Calculated
The standard equation of a circle with centre (h,k) and radius r is (x−h)² + (y−k)² = r². The general form x² + y² + 2gx + 2fy + c = 0 hides the same circle — comparing the two forms shows the centre is (−g, −f) and the radius is √(g²+f²−c). If g²+f²−c is negative, the equation doesn’t describe a real circle at all.
Worked Example
For the general equation x²+y²−4x+6y+4=0: 2g=−4 so g=−2, 2f=6 so f=3, c=4. Centre = (2,−3). Radius = √(4+9−4) = √9 = 3.
Frequently Asked Questions
Q: Why is it 2g and 2f in the general equation, not g and f?
That’s the standard convention — it’s what makes the centre come out as the clean pair (−g, −f) once you complete the square, instead of a fraction.
Q: What does it mean if the radius comes out negative under the square root?
The equation doesn’t represent a real circle in the plane — double-check the coefficients if this happens on a genuine exam question.
Converting between these two forms is a reliable Conic Sections question type in 1st PUC. See how Daniel Classes teaches 1st and 2nd PUC Mathematics →