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). If the minor axis is chosen, the surface is called an oblate spheroid (similar to the shape of the planet Earth).

Prolate spheroid.

Oblate spheroid.

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

A spheroid is a special case of an ellipsoid where two of the three major axes are equal.

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

2πb(b + a·arcsin(e)/e).

Here e is the eccentricity of the ellipse, defined as