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In geometry
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 symmetrical geometrical object. In non-mathematical usage, the term is used to refer either to a round ball or to its two-dimensional surface....
 cut off by a plane
Plane (mathematics)

In mathematics, a plane is a curvature surface. Planes can arise as subspaces of some higher dimensional space, as with the walls of a room, or they may enjoy an independent existence in their own right, as in the setting of Euclidean geometry....
. 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.

If the radius of the sphere is called , the radius of the base of the cap called , and the height of the cap called , the volume of the spherical cap is then and the curved surface area of the spherical cap is .






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In geometry
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 symmetrical geometrical object. In non-mathematical usage, the term is used to refer either to a round ball or to its two-dimensional surface....
 cut off by a plane
Plane (mathematics)

In mathematics, a plane is a curvature surface. Planes can arise as subspaces of some higher dimensional space, as with the walls of a room, or they may enjoy an independent existence in their own right, as in the setting of Euclidean geometry....
. 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.

If the radius of the sphere is called , the radius of the base of the cap called , and the height of the cap called , the volume of the spherical cap is then and the curved surface area of the spherical cap is .

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 .

Generally, the volume of a hyperspherical cap of height and radius in -dimensional Euclidean space is given by
where (the gamma function
Gamma function

In mathematics, the Gamma function is an extension of the factorial function to real number and complex number numbers. For a complex number z with positive real part the Gamma function is defined by...
) is given by .

See also

  • Circular segment
    Circular segment

    In 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 line or a chord ....
     - the analogous 2D object.
  • Dome (mathematics)
    Dome (mathematics)

    In mathematics, a dome is a closed geometry 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 joins with the rim of the disk....


External links

  • , from MathWorld
    MathWorld

    MathWorld is an online mathematics reference work, created and largely written by Eric W. Weisstein. It is sponsored by Wolfram Research Inc. and was partially funded by the National Science Foundation's National Science Digital Library grant to the University of Illinois at Urbana-Champaign....