Complete the Square for Circle Equations
Find the center (h, k) and radius r of any circle by completing the square on equations in the general form x² + y² + Dx + Ey + F = 0.
Circle Equation Calculator
Standard Form of a Circle
The standard equation of a circle is (x - h)² + (y - k)² = r², where (h, k) is the center coordinate and r is the radius. General circle equations are often given expanded, requiring completing the square twice (once for x, once for y) to find the center.
Completing the Square Twice
First, group the x terms together and the y terms together, moving the constant to the other side. Then, take half of the x-coefficient, square it, and add it to both sides. Repeat this process for the y-coefficient. Finally, factor both perfect square trinomials.
Degenerate Circles
If, after completing the square, your r² value on the right side is 0, the circle is just a single point. If it's negative, the circle is imaginary and does not exist on the real coordinate plane.
Complete the Square Calculators
Specialized tools for every completing the square scenario — pick the one that matches your problem.