Spheroid Calculator

Name: Spheroid - Geometry Calculator
Author: Jürgen Kummer

Calculations at a spheroid (ellipsoid of revolution). A spheroid is an ellipsoid with two semi axes of equal length. There are two forms: the oblate or flattened spheroid with a>c, this is the form of stars and planets. With a<c, it is a prolate or elongated spheroid, like an American football or rugby ball. With a=c, it is a sphere.
Enter the semi axes a and c, choose the number of decimal places and click Calculate.

	Equatorial semi axes (a):			Oblate spheroid Prolate spheroid
	Polar semi axis (c):
	Ellipticity (e):
	Surface area (A):
	Volume (V):
	Surface-to-volume ratio (A/V):
	Round to decimal places.

	Formulas: Oblate spheroid, a > c: $e = \sqrt{\frac{a^{2} - c^{2}}{a^{2}}}$ $A = 2 π a \cdot [a + \frac{c^{2}}{\sqrt{a^{2} - c^{2}}} \cdot arsinh (\frac{\sqrt{a^{2} - c^{2}}}{c})]$ Prolate spheroid, a < c: $e = \sqrt{\frac{c^{2} - a^{2}}{c^{2}}}$ $A = 2 π a \cdot [a + \frac{c^{2}}{\sqrt{c^{2} - a^{2}}} \cdot \arcsin (\frac{\sqrt{c^{2} - a^{2}}}{c})]$ $V = \frac{4}{3} \cdot π \cdot a^{2} \cdot c$ pi: $π = 3.141592653589793...$

The semi axes have the same unit (e.g. meter), the area has this unit squared (e.g. square meter), the volume has this unit to the power of three (e.g. cubic meter). A/V has this unit ^-1. Ellipticity is dimensionless.

The ellipticity of celestial bodies is caused by the conflict between two forces. The gravitational force pulls everything towards the center of mass, so it wants to form a sphere. The centrifugal force, which is caused by rotation, pushes outwards into a plane, so if it were only up to it, a circle would be formed. The faster a celestial body rotates around itself, the flatter it becomes. The oblate spheroid can be seen as an intermediate stage between a sphere and a circle. The smaller the ellipticity, the more spherical the spheroid is. The ellipticity of the earth is 0.081, so it is already very close to a sphere.

The ellipticity does not provide any information about whether it is a oblate or a prolate spheroid. To distinguish between them, you have to compare the lengths of the semi-axes. If you swap the values of the two different semi-axes, the oblate spheroid has a larger surface area and a larger volume than the prolate. This is because the longer axis a is present twice there and is therefore included in the weighting twice as often.

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