Ellipsoid
Calculates ellipsoid and spheroid volume and surface area.
Scalene ellipsoid
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 following formula gives the volume of an ellipsoid:
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
Spheroids
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 the formula, Earth's surface is about 510050983.92 square kilometers.
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