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
The relationship between and is irrelevant as long as . The red section of the illustration is also a spherical cap.
The parameters , and are not independent:
Substituting this into the area formula gives:
Note also that in the upper hemisphere of the diagram, , and in the lower hemisphere ; hence in either hemisphere and so an alternative expression for the volume is
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 circles of latitude
The surface area bounded by two circles of latitude is the difference of surface areas of their respective spherical caps. For a sphere of radius r, and latitudes φ1 and φ2, the area is 
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.
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 ,
- Circular segment — the analogous 2D object
- Solid angle — contains formula for n-sphere caps
- Spherical segment
- Spherical sector
- Spherical wedge
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Derivation and some additional formulas.