Spheroid Calculation

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 Delete Entries Semiaxis a Semiaxis b Decimal places 0 1 2 3 4 6 8 10  Results Volume V Surface area S

oblates ellipsoid


prolate ellipsoid

Formulas for the spheroid

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

Volume (\(\small{V}\))

\(\displaystyle V=\frac{4}{3} ·π · a·b·c\)

Surface of the oblates ellipsoid (a < b) (\(\small{S}\))

\(\displaystyle S=\frac{2\cdot π\cdot a^2\cdot b}{\sqrt{b^2-a^2}} \left[\frac{b}{a^2} \sqrt{b^2-a^2} +arcsinh\left(\frac{\sqrt{b^2-a^2}}{a} \right) \right] \)

Surface of the prolate ellipsoid (a > b) (\(\small{S}\))

\(\displaystyle S=\frac{2\cdot π\cdot a^2\cdot b}{\sqrt{a^2-b^2}} \left[\frac{b}{a^2} \sqrt{a^2-b^2} +arcsin\left(\frac{\sqrt{a^2-b^2}}{a} \right) \right] \)

Surface of a sphere (a = b) (\(\small{S}\))

\(\displaystyle S=4· a ·b·π\)