How this calculator works
Standard equations encode the centre as h and k inside squared differences. A parametric equation uses the same centre as the constant offsets of cosine and sine. General-form equations need completion of both squares. Two diameter endpoints give the centre by averaging their coordinates, and the radius is half their distance. All modes require a positive radius. The point table lists horizontal and vertical extremes in the entered coordinate unit.
Formula
General form: h=-D/2, k=-E/2, r²=h²+k²-F. Diameter endpoints: (h,k)=((x₁+x₂)/2,(y₁+y₂)/2). Parametric form: x=h+r cos(t), y=k+r sin(t).
Worked example
The equation x²+y²-6x-8y+21=0 gives centre (3,4) and radius 2. Its rightmost point is (5,4). Diameter endpoints (-1,2) and (5,10) instead give centre (2,6) and radius 5.
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Frequently asked questions
Why does (x+3)² imply a centre of -3?
Match it to (x-h)². The expression x+3 is x-(-3), so the horizontal centre is -3.
Can any two points on a circle be used?
Only diameter endpoints work in endpoint mode. A general chord does not locate a unique centre without more information.
What if cosine and sine have different amplitudes?
That generally describes an ellipse. This parametric circle mode requires the same positive amplitude r on both axes.