Spheroid

A spheroid is a quadric surface in three dimensions obtained by rotating an ellipse about one of its principal axes.
If the ellipse is rotated about its major axis, the surface is called a prolate spheroid (similar to the shape of a rugby ball or cigar).
If the minor axis is chosen, the surface is called an oblate spheroid (similar to the shape of the planet Earth).

A spheroid can also be characterised as an ellipsoid having two equal semi-axes, as represented by the equation

A prolate spheroid has one semiaxis longer than the other two, (a > b); an oblate spheroid has two equal semiaxes that are longer than the third one(a < b) and can resembles a disk.

The sphere is a special case of the spheroid in which the generating ellipse is a circle.

Volume

Prolate spheroid:

volume is
Oblate spheroid:

volume is

where

a is the major axis length

b is the minor axis length

Surface area

A prolate spheroid has surface area

An oblate spheroid has surface area

Here e is the eccentricity of the ellipse, defined as

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