In geometry, a spherical cap or spherical dome 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
If the radius of the base of the cap is , and the height of the cap is , then the volume of the spherical cap is
and the curved surface area of the spherical cap is
The relationship between and is irrelevant as long as and . The blue 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
The volume of all points which are in at least one of two intersecting spheres of radii r1 and r2 is 
is the total of the two isolated spheres, and
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
- Circular segment — the analogous 2D object.
- Dome (mathematics)
- 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