In geometry, a spherical cap, spherical dome, or spherical segment of one base is a portion of a sphere cut off by a 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.
Volume and surface area
in which is 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 it denotes the latitude in geographic coordinates, .)
The relationship between and is irrelevant as long as . For example, the red section of the illustration is also a spherical cap for which .
These formulas can be rewritten to use the radius of the base of the cap instead of , using the relationship:
Substituting this into the formulas gives:
Volumes of union and intersection of two intersecting spheres
is the sum of the volumes of the two isolated spheres, and
Surface area bounded by parallel disks
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 (north of the Arctic Circle, at latitude 66.56° as of August 2016) is 2π·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.
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 (having an axis of rotation), and likewise the ellipsoidal dome is derived from the ellipsoid.
Generally, the -dimensional volume of a hyperspherical cap of height and radius in -dimensional Euclidean space is given by 
where (the gamma function) is given by .
and the area formula can be expressed in terms of the area of the unit n-ball as
Earlier in  (1986, USSR Academ. Press) the following formulas were derived: , where ,
A more quantitive way of writing this, is in  where the bound is given. For large caps (that is when as ), the bound simplifies to .
- Circular segment — the analogous 2D object
- Solid angle — contains formula for n-sphere caps
- Spherical segment
- Spherical sector
- Spherical wedge
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|Wikimedia Commons has media related to Spherical caps.|
- Weisstein, Eric W. "Spherical cap". MathWorld. Derivation and some additional formulas.
- Online calculator for spherical cap volume and area.
- Summary of spherical formulas.