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Center-Radius Form

The center-radius form of a circle is its Cartesian equation

(x-h)^2+(y-k)^2=r^2,

where (h,k) is the center and r>0 is the radius. This form makes the center and radius explicit.

For an equation x^2+y^2+Dx+Ey+F=0, completing the square gives

(x+D/2)^2+(y+E/2)^2=(D^2+E^2)/4-F.

The equation represents a circle when the right side is positive, a point circle when it is zero, and no points with real coordinates when it is negative (Story).

See also

Cartesian Equation, Circle, Completing the Square, Radius

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References

Story, D. P. "Some Second Degree & Trig Curves." Lesson 10 in Algebra Review in Ten Lessons. https://web.archive.org/web/20091228232155id_/http://www.math.uakron.edu/~dpstory/tutorial/mptii/lesson10.pdf.

Cite this as:

Weisstein, Eric W. "Center-Radius Form." From MathWorld--A Wolfram Resource. https://mathworld.wolfram.com/Center-RadiusForm.html

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