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, so that the height of the cap is equal to the radius of the sphere, the spherical cap is called a hemisphere.
The polar angle between the rays from the center of the sphere to the apex of the cap and the edge of the disk forming the base of the cap
Using and
Using and
Using and
Volume
Area
If denotes the latitude in geographic coordinates, then. 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 volume formula of, where is the infinitesimal area of each pyramidal base and is the height of each pyramid from its base to its apex. 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:
Deriving the volume and surface area using calculus
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. The area is The derivative of is and hence The formula for the area is therefore The volume is
Applications
Volumes of union and intersection of two intersecting spheres
The volume of the union of two intersecting spheres of radii and is where is the sum of the volumes of the two isolated spheres, and the sum of the volumes of the two spherical caps forming their intersection. If is the distance between the two sphere centers, elimination of the variables and leads to
Areas of intersecting spheres
Consider two intersecting spheres of radii and, with their centers separated by distance. They intersect if From the law of cosines, the polar angle of the spherical cap on the sphere of radius is Using this, the surface area of the spherical cap on the sphere of radius is
The curved surface area of the spherical segment bounded by two parallel disks is the difference of surface areas of their respective spherical caps. For a sphere of radius, and caps with heights and, the area is or, using geographic coordinates with latitudes and, For example, assuming the Earth is a sphere of radius 6371 km, the surface area of the arctic is 2·63712|sin 90° − sin 66.56°| = 21.04 million km2, or 0.5·|sin 90° − sin 66.56°| = 4.125% of the total surface area of the Earth. This formula can also be used to demonstrate that half the surface area of the Earth lies between latitudes 30° South and 30° North in a spherical zone which encompasses all of the Tropics.
Generalizations
Sections of other solids
The spheroidal dome is obtained by sectioning off a portion of a spheroid so that the resulting dome is circularly symmetric, and likewise the ellipsoidal dome is derived from the ellipsoid.
Hyperspherical cap
Generally, the -dimensional volume of a hyperspherical cap of height and radius in -dimensional Euclidean space is given by: where is given by. The formula for can be expressed in terms of the volume of the unitn-ball and the hypergeometric function or the regularized incomplete beta function as and the area formula can be expressed in terms of the area of the unit n-ball as where. Earlier in the following formulas were derived: , where For odd
Asymptotics
It is shown in that, if and, then where is the integral of the standard normal distribution. A more quantitative bound is For large caps, the bound simplifies to.
