Spherical cap

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 These variables are inter-related through the formulas

denotes the latitude in geographic coordinates, then

cos ⁡ θ = sin ⁡ ϕ

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,[2] 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

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).

, 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.

is [3] 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.

is the distance between the two sphere centers, elimination of the variables

leads to[4][5] The volume of a spherical cap with a curved base can be calculated by considering two spheres with radii

The volume is thus the difference between sphere 2's cap (with height

This formula is valid only for configurations that satisfy

, which is the case for a spherical cap with a base that has a negligible curvature, the above equation is equal to the volume of a spherical cap with a flat base, as expected.

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.

, the area is or, using geographic coordinates with latitudes

,[6] For example, assuming the Earth is a sphere of radius 6371 km, the surface area of the arctic (north of the Arctic Circle, at latitude 66.56° as of August 2016[7]) is 2π ⋅ 63712 |sin 90° − sin 66.56°| = 21.04 million km2 (8.12 million sq mi), 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.

The spheroidal dome is obtained by sectioning off a portion of a spheroid so that the resulting dome is circularly symmetric (having an axis of rotation), and likewise the ellipsoidal dome is derived from the ellipsoid.

-dimensional volume of a hyperspherical cap of height

can be expressed in terms of the volume of the unit n-ball

can be expressed in terms of the area of the unit n-ball