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Concept
Spherical cap
Summary
In geometry, a spherical cap or spherical dome is a portion of a sphere or of a ball cut off by a plane. It is also a spherical segment of one base, i.e., bounded by a single plane. If the plane passes through the center of the sphere (forming a great circle), so that the height of the cap is equal to the radius of the sphere, the spherical cap is called a hemisphere.
The volume of the spherical cap and the area of the curved surface may be calculated using combinations of
The radius of the sphere
The radius of the base of the cap
The height of the cap
The polar angle between the rays from the center of the sphere to the apex of the cap (the pole) and the edge of the disk forming the base of the cap
If denotes the latitude in geographic coordinates, then , and .
The relationship between and is relevant as long as . For example, the red section of the illustration is also a spherical cap for which .
The formulas using and can be rewritten to use the radius of the base of the cap instead of , using the Pythagorean theorem:
so that
Substituting this into the formulas gives:
Note that aside from the calculus based argument below, the area of the spherical cap may be derived from the volume of the spherical sector, by an intuitive argument, as
The intuitive argument is based upon summing the total sector volume from that of infinitesimal triangular pyramids. Utilizing the pyramid (or cone) volume formula of , where is the infinitesimal area of each pyramidal base (located on the surface of the sphere) and is the height of each pyramid from its base to its apex (at the center of the sphere). Since each , in the limit, is constant and equivalent to the radius of the sphere, the sum of the infinitesimal pyramidal bases would equal the area of the spherical sector, and:
The volume and area formulas may be derived by examining the rotation of the function
for , using the formulas the surface of the rotation for the area and the solid of the revolution for the volume.
Official source
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