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This equation reduces to that of the volume of a sphere when all three elliptic radii are equal, and to that of an oblate or prolate spheroid when two of them are equal. The volume of an ellipsoid is 2 / 3 the volume of a circumscribed elliptic cylinder, and π / 6 the volume of the circumscribed box.
The area formula is intuitive: start with a circle of radius (so its area is ) and stretch it by a factor / to make an ellipse. This scales the area by the same factor: π b 2 ( a / b ) = π a b . {\displaystyle \pi b^{2}(a/b)=\pi ab.} [ 18 ] However, using the same approach for the circumference would be fallacious – compare the integrals
A sphere (top), rotational ellipsoid (left) and triaxial ellipsoid (right) The volume of a sphere of radius R is .Given the volume of a non-spherical object V, one can calculate its volume-equivalent radius by setting
Area#Area formulas – Size of a two-dimensional surface; Perimeter#Formulas – Path that surrounds an area; List of second moments of area; List of surface-area-to-volume ratios – Surface area per unit volume; List of surface area formulas – Measure of a two-dimensional surface; List of trigonometric identities
An ellipse has two axes and two foci Unlike most other elementary shapes, such as the circle and square , there is no algebraic equation to determine the perimeter of an ellipse . Throughout history, a large number of equations for approximations and estimates have been made for the perimeter of an ellipse.
If the ellipse is rotated about its major axis, the result is a prolate spheroid, elongated like a rugby ball. The American football is similar but has a pointier end than a spheroid could. If the ellipse is rotated about its minor axis, the result is an oblate spheroid, flattened like a lentil or a plain M&M.
A circle of radius a compressed to an ellipse. A sphere of radius a compressed to an oblate ellipsoid of revolution. Flattening is a measure of the compression of a circle or sphere along a diameter to form an ellipse or an ellipsoid of revolution respectively. Other terms used are ellipticity, or oblateness.
More formulas of this nature can be given, as explained by Ramanujan's theory of elliptic functions to alternative bases. Perhaps the most notable hypergeometric inversions are the following two examples, involving the Ramanujan tau function τ {\displaystyle \tau } and the Fourier coefficients j {\displaystyle \mathrm {j} } of the J-invariant ...