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A ball in n dimensions is called a hyperball or n-ball and is bounded by a hypersphere or (n−1)-sphere. Thus, for example, a ball in the Euclidean plane is the same thing as a disk, the area bounded by a circle. In Euclidean 3-space, a ball is taken to be the volume bounded by a 2-dimensional sphere. In a one-dimensional space, a ball is a ...
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 − 1)-sphere of radius r, then: = (). Applying this to the above integral gives the expression
An open ball excludes the sphere itself, while a closed ball includes the sphere: a closed ball is the union of the open ball and the sphere, and a sphere is the boundary of a (closed or open) ball. The distinction between ball and sphere has not always been maintained and especially older mathematical references talk about a sphere as a solid.
Let be the surface area of the unit -sphere of radius embedded in -dimensional Euclidean space, and let be the volume of its interior, the unit -ball. The surface area of an arbitrary -sphere is proportional to the st power of the radius, and the volume of an arbitrary -ball is ...
For example, assuming the Earth is a sphere of radius 6371 km, the surface area of the arctic (north of the Arctic Circle, at latitude 66.56° as of August 2016 [7]) is 2π ⋅ 6371 2 | sin 90° − sin 66.56° | = 21.04 million km 2 (8.12 million sq mi), or 0.5 ⋅ | sin 90° − sin 66.56° | = 4.125% of the total surface area of the Earth ...
Plot of the surface-area:volume ratio (SA:V) for a 3-dimensional ball, showing the ratio decline inversely as the radius of the ball increases. A solid sphere or ball is a three-dimensional object, being the solid figure bounded by a sphere. (In geometry, the term sphere properly refers only to the surface, so a sphere thus lacks volume in this ...
The dimensionless quantities often represent the degree of deviation from an ideal shape, such as a circle, sphere or equilateral polyhedron. [1] Shape factors are often normalized, that is, the value ranges from zero to one. A shape factor equal to one usually represents an ideal case or maximum symmetry, such as a circle, sphere, square or cube.
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 .