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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 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 the n-ball () can be computed by integrating the volume element in spherical coordinates. The spherical coordinate system has a radial coordinate r and angular coordinates φ 1, …, φ n − 1, where the domain of each φ except φ n − 1 is [0, π), and the domain of φ n − 1 is [0, 2 π). The spherical volume element is:
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 : (), = ….
The formula for the volume of the -ball can be derived from this by integration. Similarly the surface area element of the ( n − 1 ) {\displaystyle (n-1)} -sphere of radius r {\displaystyle r} , which generalizes the area element of the 2 {\displaystyle 2} -sphere, is given by
The surface of the spherical segment (excluding the bases) is called spherical zone. Geometric parameters for spherical segment. If the radius of the sphere is called R , the radii of the spherical segment bases are a and b , and the height of the segment (the distance from one parallel plane to the other) called h , then the volume of the ...
Many problems in the chemical and physical sciences can be related to packing problems where more than one size of sphere is available. Here there is a choice between separating the spheres into regions of close-packed equal spheres, or combining the multiple sizes of spheres into a compound or interstitial packing.
If the radius of the sphere is denoted by r and the height of the cap by h, the volume of the spherical sector is =. This may also be written as V = 2 π r 3 3 ( 1 − cos φ ) , {\displaystyle V={\frac {2\pi r^{3}}{3}}(1-\cos \varphi )\,,} where φ is half the cone aperture angle, i.e., φ is the angle between the rim of the cap and the ...