Prolate.html

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

A prolate spheroid

1 Prolate spheroids in sport
2 Prolate spheroids in astronomy
3 Surface area
4 Volume
5 See also
6 References

Prolate spheroids in sport

The prolate spheroid is the shape of the ball in several sports, such as Rugby Football and Australian Rules Football. American Football and Canadian Football use a pointed prolate spheroid (also resembling a rotated vesica piscis).¹

Prolate spheroids in astronomy

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 (moons of Saturn) and Miranda (moon of Uranus). Also, the dwarf planet Haumea is a prolate spheroid.

It is also used to describe the shape of some nebulae such as the Crab Nebula.²

Surface area

A prolate spheroid has surface area

$2\pi\left(a^2+\frac{a b o\!\varepsilon}{\sin(o\!\varepsilon)}\right)$

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