Spheroid
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In mathematics, 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 or a pancake).
A spheroid can also be characterised as an ellipsoid having two equal semi-axes (e.g. b = c), as represented by the equation
- [\frac+\frac+\frac=1]
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The sphere is a special case of the spheroid in which the generating ellipse is a circle.
Volume
Prolate spheroid:
- volume is [\frac\pi a b^2]
- volume is [\frac\pi a^2 b]
- a is the semi-major axis length
- b is the semi-minor axis length
Surface area
A prolate spheroid has surface area
- [2\pi b\left(b + a \frac}\right).\,\!]
- [\pi\left(2 a^2 + \frac \log\left(\frac\right) \right).\,\!]
- a is the semi-major axis length
- b is the semi-minor axis length
- e is the eccentricity of the ellipse
- :[\begin\\_\\e&=&\!\!\!\sin(O\!\!E),\qquad\qquad\qquad\qquad\qquad\qquad\qquad\quad\\&&\!\!\!\!\!\!\!\!\!\!\!\!^O\!\!E\mathrm\mathit\mathrm\mathit)}\\&=&\!\!\!\!\!\!\!\!\!\sqrt}\quad\mathrm,\qquad\qquad\qquad\qquad\quad\\\\&=&\!\!\!\!\!\!\!\sqrt}\quad\mathrm.\qquad\qquad\qquad\qquad\quad\\^\end\,\!]
See also
External links
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