 
oblate spheroid
 prolate spheroid

A spheroid is a quadric surfaceobtained by rotating an ellipse about one of its principal axes; in other words, an ellipsoid with two equal semidiameters.
If the ellipse is rotated about its major axis, the result is a prolate (elongated) spheroid, somewhat similar to a rugby ball. If the ellipse is rotated about its minor axis, the result is an oblate (flattened) spheroid, somewhat similar to a lentil. If the generating ellipse is a circle, the surface is a sphere.
Because of its rotation, the Earth's shape is more similar to an oblate spheroid with a ≈ 6,378.137 km and b ≈ 6,356.752 km, than to a sphere.
Equation
A spheroid centered at the origin and rotated about the
z axis is defined by the
implicit equation
 $left(frac\{x\}\{a\}right)^2+left(frac\{y\}\{a\}right)^2+left(frac\{z\}\{b\}right)^2\; =\; 1quadquadhbox\{\; or\; \}quadquadfrac\{x^2+y^2\}\{a^2\}+frac\{z^2\}\{b^2\}=1$
where
a is the horizontal, transverse radius at the equator, and
b is the vertical, conjugate radius.
Surface area
A prolate spheroid has
surface area
 $2pileft(a^2+frac\{a\; b\; o!varepsilon\}\{sin(o!varepsilon)\}right)$
where
$o!varepsilon=arccosleft(frac\{a\}\{b\}right)$ is the
angular eccentricity of the ellipse, and
$e=sin(o!varepsilon)$ is its (ordinary)
eccentricity.
An oblate spheroid has surface area
 $2pileft[a^2+frac\{b^2\}\{sin(o!varepsilon)\}\; lnleft(frac\{1+\; sin(o!varepsilon)\}\{cos(o!varepsilon)\}right)right]$.
Volume
The volume of a spheroid (of any kind) is
$frac\{4\}\{3\}pi\; a^2b.$
Curvature
If a spheroid is parameterized as
 $vec\; sigma\; (beta,lambda)\; =\; (a\; cos\; beta\; cos\; lambda,\; a\; cos\; beta\; sin\; lambda,\; b\; sin\; beta);,!$
where
$beta,!$ is the
reduced or
parametric latitude,
$lambda,!$ is the
longitude, and
$frac\{pi\}\{2\}+frac\{pi\}\{2\},!\; math>\; and$ pi\mapsto +pi,!\; math>,\; then\; itsGaussian\; curvatureis$+pi,!>$ $K(beta,lambda)\; =\; \{b^2\; over\; (a^2\; +\; (b^2\; \; a^2)\; cos^2\; beta)^2\};,!$
and its
mean curvature is
 $H(beta,lambda)\; =\; \{b\; (2\; a^2\; +\; (b^2\; \; a^2)\; cos^2\; beta)\; over\; 2\; a\; (a^2\; +\; (b^2\; \; a^2)\; cos^2\; beta)^\{3/2\}\}.,!$
Both of these curvatures are always positive, so that every point on a spheroid is elliptic.
See also
External links