Spherical cap
Encyclopedia
Geometry
Geometry arose as the field of knowledge dealing with spatial relationships. Geometry was one of the two fields of pre-modern mathematics, the other being the study of numbers ....
, a spherical cap is a portion of a sphere
Sphere
A sphere is a perfectly round geometrical object in three-dimensional space, such as the shape of a round ball. Like a circle in two dimensions, a perfect sphere is completely symmetrical around its center, with all points on the surface lying the same distance r from the center point...
cut off by a plane
Plane (mathematics)
In mathematics, a plane is a flat, two-dimensional surface. A plane is the two dimensional analogue of a point , a line and a space...
. If the plane passes through the center of the sphere, so that the height of the cap is equal to the radius
Radius
In classical geometry, a radius of a circle or sphere is any line segment from its center to its perimeter. By extension, the radius of a circle or sphere is the length of any such segment, which is half the diameter. If the object does not have an obvious center, the term may refer to its...
of the sphere, the spherical cap is called a hemisphere.
If the radius of the sphere is
, the radius of the base of the cap is 
, and the height of the cap is 
, then the volumeVolume
Volume is the quantity of three-dimensional space enclosed by some closed boundary, for example, the space that a substance or shape occupies or contains....
of the spherical cap is
and the curved surface area
Area
Area is a quantity that expresses the extent of a two-dimensional surface or shape in the plane. Area can be understood as the amount of material with a given thickness that would be necessary to fashion a model of the shape, or the amount of paint necessary to cover the surface with a single coat...
of the spherical cap is
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
.Hyperspherical cap
Generally, the
-dimensional volume of a hyperspherical cap of height 
and radius 
in 
-dimensional Euclidean space is given by
where
(the gamma functionGamma function
In mathematics, the gamma function is an extension of the factorial function, with its argument shifted down by 1, to real and complex numbers...
) is given by
.The formula for
can be expressed in terms of the volume of the unit n-ballN-ball
N-ball is a two-dimensional physics game released by Rag Doll Software in 2005. The game is divided into four different campaigns with a total of 40 levels. The central idea of the game is move the glowing ball to the finish plate. This is done by holding or tapping the space bar to make the 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-ballN-ball
N-ball is a two-dimensional physics game released by Rag Doll Software in 2005. The game is divided into four different campaigns with a total of 40 levels. The central idea of the game is move the glowing ball to the finish plate. This is done by holding or tapping the space bar to make the ball...
as
,where
.See also
- Circular segmentCircular segmentIn geometry, a circular segment is an area of a circle informally defined as an area which is "cut off" from the rest of the circle by a secant or a chord. The circle segment constitutes the part between the secant and an arc, excluding the circle's center...
— the analogous 2D object. - Dome (mathematics)Dome (mathematics)In mathematics, a dome is a closed geometrical surface which can be obtained by sectioning off a portion of a sphere with an intersecting plane. It consists of two parts: a flat disk, which is joined to a convex surface whose curvature is uniform and which has a circular boundary: this boundary...
- Solid angleSolid angleThe solid angle, Ω, is the two-dimensional angle in three-dimensional space that an object subtends at a point. It is a measure of how large that object appears to an observer looking from that point...
— contains formula for n-sphere caps - Spherical segmentSpherical segmentIn geometry, a spherical segment is the solid defined by cutting a sphere with a pair of parallel planes.It can be thought of as a spherical cap with the top truncated, and so it corresponds to a spherical frustum...
- Spherical sectorSpherical sectorIn geometry, a spherical sector is a portion of a sphere defined by a conical boundary with apex at the center of the sphere. It can be described as the union of a spherical cap and the cone formed by the center of the sphere and the base of the cap....
- Spherical wedgeSpherical wedgeIn geometry, a spherical wedge or ungula is a portion of a ball bounded by two plane semidisks and a spherical lune . The angle between the radii lying within the bounding semidisks is the dihedral angle of the wedge α...