Ellipsoid is a sphere-like surface for which all cross-sections are ellipses.

Equation of standard ellipsoid body in xyz coordinate system is

,

where a - radius along x axis, b - radius along y axis, c - radius along z axis.

The volume of an ellipsoid is given by the following formula:

The surface area of a general ellipsoid cannot be expressed exactly by an elementary function. Knud Thomsen from Denmark proposed the following approximate formula: , where p=1.6075

If any two of the three axes of an ellipsoid are equal, the figure becomes a spheroid (ellipsoid of revolution). There are two kinds of spheroid: oblate spheroid (lens like) and prolate spheroid (сigar like).

Volume of spheroid is calculated by the following formula:

Unlike ellipsoids, exact surface area formulas exist for spheroids:

For oblate spheroid (a = b > c):

where angular eccentricity

For prolate spheroid (a = b < c):

where angular eccentricity

The Earth's shape is similar to an oblate spheroid with a ≈ 6,378.137 km and c ≈ 6,356.752 km. According to formula, Earth's surface is about 510050983.92 square kilometers.

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