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An example of a spherical cap in blue (and another in red) In geometry, a spherical cap or spherical dome is a portion of a sphere or of a ball cut off by a plane.It is also a spherical segment of one base, i.e., bounded by a single plane.
The volume of a n-ball is the Lebesgue measure of this ball, which generalizes to any dimension the usual volume of a ball in 3-dimensional space. The volume of a n -ball of radius R is R n V n , {\displaystyle R^{n}V_{n},} where V n {\displaystyle V_{n}} is the volume of the unit n -ball , the n -ball of radius 1 .
The ratio of the volume of a sphere to the volume of its circumscribed cylinder is 2:3, as was determined by Archimedes. The principal formulae derived in On the Sphere and Cylinder are those mentioned above: the surface area of the sphere, the volume of the contained ball, and surface area and volume of the cylinder.
For most practical purposes, the volume inside a sphere inscribed in a cube can be approximated as 52.4% of the volume of the cube, since V = π / 6 d 3, where d is the diameter of the sphere and also the length of a side of the cube and π / 6 ≈ 0.5236.
Solid hemisphere: r = the radius of the hemisphere Solid semi-ellipsoid of revolution around z-axis: a = the radius of the base circle h = the height of the semi-ellipsoid from the base cicle's center to the edge
Consider the linear subspace of the n-dimensional Euclidean space R n that is spanned by a collection of linearly independent vectors , …,. To find the volume element of the subspace, it is useful to know the fact from linear algebra that the volume of the parallelepiped spanned by the is the square root of the determinant of the Gramian matrix of the : (), = ….
Hart (2009) [3] states that the "volume of a spherical wedge is to the volume of the sphere as the number of degrees in the [angle of the wedge] is to 360". Hence, and through derivation of the spherical wedge volume formula, it can be concluded that, if V s is the volume of the sphere and V w is the volume of a given spherical wedge,
An approximation for the volume of a thin spherical shell is the surface area of the inner sphere multiplied by the thickness t of the shell: [2], when t is very small compared to r (). The total surface area of the spherical shell is .