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# Formulas and Tables

Elementary Geometry

# Ellipsoid

Semi-axes of an ellipsoid: $$a$$, $$b$$, $$c$$
Semi-axes of a spheroid: $$a$$, $$b$$, $$b$$
Eccentricity: $$e$$

Surface area: $$S$$
Volume: $$V$$

1. An ellipsoid is a closed surface in space (as well as a geometric solid bounded by this surface), which is formed as a result of deformation of a sphere along the three orthogonal coordinate axes. An ellipsoid is a three-dimensional analogue of an ellipse and is defined by three semi-axes $$a$$, $$b$$, $$c.$$
1. Volume of an ellipsoid
$$V = {\large\frac{{4\pi abc}}{3}\normalsize}$$
2. 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 $$a$$, $$b$$, $$b$$ ($$a \lt b$$).
3. Surface area of a prolate spheroid
$$S =$$ $$2\pi b\left( {b + {\large\frac{{a\arcsin e}}{e}}\normalsize} \right),$$ where $$e = {\large\frac{{\sqrt {{a^2} – {b^2}} }}{a}\normalsize},$$ $$\left( {a \gt b} \right)$$
4. 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 $$e = {\large\frac{{\sqrt {{b^2} – {a^2}} }}{a}\normalsize},$$ $$\left( {a \lt b} \right)$$
5. Volume of a spheroid
$$V = {\large\frac{{4\pi a{b^2}}}{3}\normalsize}$$