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For example, a sphere with diameter 1 m has 52.4% the volume of a cube with edge length 1 ... A sphere of radius r has area element = ...
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 ...
The 3-sphere is the boundary of a -ball in four-dimensional space. The -sphere is the boundary of an -ball. Given a Cartesian coordinate system, the unit -sphere of radius can be defined as:
Let (M, d) be a metric space, namely a set M with a metric (distance function) d, and let be a positive real number. The open (metric) ball of radius r centered at a point p in M , usually denoted by B r ( p ) or B ( p ; r ) , is defined the same way as a Euclidean ball, as the set of points in M of distance less than r away from p , B ...
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.
Thus, the segment volume equals the sum of three volumes: two right circular cylinders one of radius a and the second of radius b (both of height /) and a sphere of radius /. The curved surface area of the spherical zone—which excludes the top and bottom bases—is given by =.
A sphere (top), rotational ellipsoid (left) and triaxial ellipsoid (right) The volume of a sphere of radius R is . Given the volume of a non-spherical object V, one can calculate its volume-equivalent radius by setting = or, alternatively:
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 angle, i.e., φ is the angle between the rim of the cap and the direction ...
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