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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 or a Pancake ).

A spheroid can also be characterised as an Ellipsoid having two equal Semi-axes (''b'' = ''c''), as represented by the equation

: rac{x^2}{a^2}+ rac{y^2}{b^2}+ rac{z^2}{b^2}=1

A prolate spheroid has a semi-minor axis shorter than the semi-major axis (''a'' > ''b''); an oblate spheroid has a semi-minor axis longer than the semi-major axis (''a'' < ''b'') and can resemble a Disk .




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


VOLUME


Prolate spheroid:
  • volume is rac{4}{3}\pi a b^2

    • Oblate spheroid:

  • volume is rac{4}{3}\pi a^2 b


where
  • ''a'' is the semi-major axis length

  • ''b'' is the semi-minor axis length



SURFACE AREA


An oblate spheroid has Surface Area

:\pi\left(2 a^2 + rac{b^2}{e} \log\left( rac{1+e}{1-e} ight) ight).

A prolate spheroid has surface area

:2\pi b\left(b + a rac{\arcsin{e}}{e} ight).

Here ''e'' is the Eccentricity of the ellipse, defined as

:e=\sqrt{1- rac{b^2}{a^2}}.

where
  • ''a'' is the semi-major axis length

  • ''b'' is the semi-minor axis length



See also