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A closed n-ball of radius r is the set of all points of distance less than or equal to r away from x. In Euclidean n -space, every ball is bounded by a hypersphere . The ball is a bounded interval when n = 1 , is a disk bounded by a circle when n = 2 , and is bounded by a sphere when n = 3 .
Sphericity is a measure of how closely the shape of an object resembles that of a perfect sphere. For example, the sphericity of the balls inside a ball bearing determines the quality of the bearing, such as the load it can bear or the speed at which it can turn without failing. Sphericity is a specific example of a compactness measure of a shape.
S 1: a 1-sphere is a circle of radius r; S 2: a 2-sphere is an ordinary sphere; S 3: a 3-sphere is a sphere in 4-dimensional Euclidean space. Spheres for n > 2 are sometimes called hyperspheres. The n-sphere of unit radius centered at the origin is denoted S n and is often referred to as "the" n-sphere. The ordinary sphere is a ...
where S n − 1 (r) is an (n − 1)-sphere of radius r (being the surface of an n-ball of radius r) and dA is the area element (equivalently, the (n − 1)-dimensional volume element). The surface area of the sphere satisfies a proportionality equation similar to the one for the volume of a ball: If A n − 1 ( r ) is the surface area of an ( n ...
Lines, L. (1965), Solid geometry: With Chapters on Space-lattices, Sphere-packs and Crystals, Dover. Reprint of 1935 edition. A problem on page 101 describes the shape formed by a sphere with a cylinder removed as a "napkin ring" and asks for a proof that the volume is the same as that of a sphere with diameter equal to the length of the hole.
A sphere (top), rotational ellipsoid (left) and triaxial ellipsoid (right) The volume of a sphere of radius R is .Given the volume of an non-spherical object V, one can calculate its volume-equivalent radius by setting
A sphere of radius r has surface area 4πr 2.. The surface area (symbol A) of a solid object is a measure of the total area that the surface of the object occupies. [1] The mathematical definition of surface area in the presence of curved surfaces is considerably more involved than the definition of arc length of one-dimensional curves, or of the surface area for polyhedra (i.e., objects with ...
a 1-sphere is a circle of radius centered at , and is the boundary of a disk ( -ball). a 2 -sphere is an ordinary 2 {\displaystyle 2} -dimensional sphere in 3 {\displaystyle 3} -dimensional Euclidean space, and is the boundary of an ordinary ball ( 3 {\displaystyle 3} -ball).