# Formulas and Tables

Elementary Geometry# Ellipsoid

Semi-axes of a spheroid: \(a\), \(b\), \(b\)

Eccentricity: \(e\)

Surface area: \(S\)

Volume: \(V\)

- Volume of an ellipsoid

\(V = {\large\frac{{4\pi abc}}{3}\normalsize}\) - A particular case of ellipsoid is a spheroid or an ellipsoid of revolution. A spheroid is formed by rotating an ellipse about one of its axes. If the ellipse is rotated about the major axis, a prolate spheroid with the semi-axes \(a\), \(b\), \(b\) (\(a \gt b\)) is obtained. If \(a\) is the minor axis of the ellipse, the result is an oblate spheroid with the semi-axes \(a\), \(b\), \(b\) (\(a \lt b\)).
- Surface area of a prolate spheroid

\(S =\) \(2\pi b\left( {b + {\large\frac{{a\arcsin e}}{e}}\normalsize} \right),\) where \(e = {\large\frac{{\sqrt {{a^2} – {b^2}} }}{a}\normalsize},\) \(\left( {a \gt b} \right)\) - Surface area of an oblate spheroid

\(S =\) \(2\pi b\left( {b + {\large\frac{{a\,{\text{arcsinh }} {\large\frac{{be}}{a}}\normalsize}}{{\large\frac{{be}}{a}}\normalsize}}\normalsize} \right),\) where \(e = {\large\frac{{\sqrt {{b^2} – {a^2}} }}{a}\normalsize},\) \(\left( {a \lt b} \right)\) - Volume of a spheroid

\(V = {\large\frac{{4\pi a{b^2}}}{3}\normalsize}\)