How this calculator works
A sphere consists of points a fixed distance from its centre. Completing squares converts the expanded equation to centre-radius form. Diameter endpoints give the centre by averaging and radius by half their distance. A zero radius is accepted as a degenerate point sphere. Coordinate-slice modes return two surface values, one at tangency, or an error when the fixed coordinates miss the sphere. Geometry is Euclidean and all coordinates use one common unit.
Formula
(x-h)^2+(y-k)^2+(z-l)^2=r^2. E=-2h, F=-2k, G=-2l, H=h^2+k^2+l^2-r^2. Surface area=4pi r^2; volume=4pi r^3/3.
Worked example
For centre (1, 2, 3) and radius 2, the equation is (x-1)^2+(y-2)^2+(z-3)^2=4. Expanded coefficients are E=-2, F=-4, G=-6, H=10.
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Frequently asked questions
What if radius squared is negative?
No real sphere exists, so the expanded equation is rejected.
Why can a surface coordinate have two answers?
A line parallel to one coordinate axis can pass through both sides of the sphere.
Can this describe an ellipsoid?
No. The squared x, y and z coefficients are all one. Unequal coefficients require an ellipsoid model.