N-Dimensional Sphere (Hypersphere) Volume and Surface Area

Compute the volume and surface area of an n-dimensional sphere (hypersphere) from the dimension n and radius r using the gamma function.

Input

Enter the dimension n and radius r to compute the volume and surface area of an n-dimensional sphere (hypersphere) using the gamma function.

Dimension n

Radius r

Volume

523.598776

Surface area

314.159265

Dimension n

Radius r

The length unit matches your input. The n-dimensional volume is in that unit to the power n, and the surface area to the power n minus 1.

The volume is V＝π^(n/2)／Γ(n/2＋1)·r^n, where Γ is the gamma function, a generalization of the factorial to non-integer arguments.
The surface area, the n−1 dimensional measure of the hypersphere boundary, is S＝n·V／r. For n＝2 the area is πr², and for n＝3 the volume is (4/3)πr³.
The gamma function is evaluated through a log-gamma approximation (Lanczos) and exponentiated, which avoids overflow and keeps results stable in high dimensions.
Enter a dimension n that is a positive integer and a positive radius r. The length unit matches the input, and the n-dimensional volume is in that unit raised to the power n.

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