A prolate spheroid is a spheroid in which the polar diameter is greater than the equatorial diameter.

Properties

A prolate spheroid has the surface area

where $o\backslash !\backslash varepsilon=\backslash arccos\backslash left(\backslash frac\{a\}\{b\}\backslash right)$ is the angular eccentricity of the ellipse, $e=\backslash sin(o\backslash !\backslash varepsilon)$ is its (ordinary) eccentricity, $\backslash \; b$ is the polar radius, and $\backslash \; a$ is the equatorial radius.

The volume of a prolate spheroid is calculated by $V\; =\; \backslash frac\{4\}\{3\}\backslash pi\; a^2\; b$

Uses

The prolate spheroid is the shape of the ball in several sports, such as in Rugby league, Rugby union and Australian Rules Football. In American Football and Canadian Football, a more pointed prolate spheroid is used (one resembling a rotated vesica piscis).

The prolate spheroid, like its opposite, the oblate spheroid, is the shape of some of the moons in the solar system. Examples are Mimas, Enceladus, and Tethys (satellites of Saturn) and Miranda (a satellite of Uranus). The dwarf planet is a scalene ellipsoid.

It is also used to describe the shape of some nebulae (nebulas) such as the Crab Nebula.
The most common shapes for the density distribution of protons and neutrons in an atomic nucleus are spherical, prolate and oblate spheroidal. Deformed shapes occur as a result of the competition between electromagnetic repulsion between protons, surface tension and quantum shell effects.

See also

• Equidimensional

• List of solar system objects by mass

• Oblate

• Polar bulge

• Reference ellipsoid

• Spheroid

• Triaxial