Spheroid Calculation

Calculator and formulas for calculating a spheroid (ellipsoid of revolution)

Spheroid caltulation


This function calculates the volume and the surface area of a spheroid. A spheroid (ellipsoid of revolution) is an elliptical body, as it arises from the rotation of an ellipse around the axis a. In contrast to a three-axis ellipsoid, axes b and c are the same length.

A distinction is made between:

  • the oblate ellipsoid, a < b, c (shape of a lens)
  • the prolate ellipsoid, a > b, c (shape of rugby ball)

To calculate the spheroid, enter the lengths of the two semiaxes a and b. Then click the 'Calculate' button.


Spheroid calculator

 Input
Semiaxis a
Semiaxis b
Decimal places
 Results
Volume V
Surface area S
RotationEllipsoid
oblates ellipsoid

RotationEllipsoid
prolate ellipsoid

Formulas for the spheroid

To calculate the surface, apply to oblates and prolate ellipsoids different formulas.

Volume
\(\displaystyle V=\frac{4}{3} ·π · a·b·c\)
 
Surface of the oblates ellipsoid (a < b)
\(\displaystyle S=\frac{2πa^2b}{\sqrt{b^2-a^2}} \left[\frac{b}{a^2} \sqrt{b^2-a^2} +arcsinh\left(\frac{\sqrt{b^2-a^2}}{a} \right) \right] \)
 
Surface of the prolate ellipsoid (a > b)
\(\displaystyle S=\frac{2πa^2b}{\sqrt{a^2-b^2}} \left[\frac{b}{a^2} \sqrt{a^2-b^2} +arcsin\left(\frac{\sqrt{a^2-b^2}}{a} \right) \right] \)
 
Surface of a sphere (a = b)
\(\displaystyle S=4· a ·b·π\)

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