Volume and Surface of the Hypersphere

This paper presents a simply method to calculate the volume and the surface of a hypersphere, and is mainly addressed to students who study physics or engineering. The basic knowledges of geometry and mathematical analysis are sufficient to understand the material presented below


Introduction
The content, or the generalized volume of a domain(body) in a Euclidean n-dimensional space, shows the amount of space occupied by this domain.For example, if the domain is a curve (one-dimensional), then the content is expressed by its length.The content of a surface (two-dimensional) is its area.For the three-dimensional subspaces the content is identical with the classical volume.
The generalized surface(frontier), or boundary of a body in n dimensional Euclidian space represents the (n-1)-dimensional domain, which separates the body from the rest of space.Let us see as example the two-dimensional sphere (circle).See the figure 1, below: The differential volume is a rectangle with the length of 2Rcosβ, and width of dx 2 (the differential of the variable x 2 ).
(1.4) The expression V 2 , which represents the "volume" for a unit radius, is a numerical constant.We may write: (1.5) The S 2 is the perimeter of the circle which separates the kernel of the "two-dimensional sphere" from the neighboring space.

The Volume of a N-Dimensional Sphere
The coordinates of a point P in an n-dimensional cartesian system can be written as: It is obvious that all points with the same radius(R) build a n-dimensional sphere in the corresponding space.Its content takes, generally speaking, the following form: There is a network of (n-1)-dimensional spheres each having a radius of Rcosβ, which belongs to this n-dimensional sphere.It is like this n-dimensional sphere is built of slices with "bases" consisting of such (n-1)-dimensional spheres.

The Surface of the N-Dimensional Hypersphere
As was mentioned in the introduction, a hyper-sphere is separated from the complementary n-dimensional space by a boundary, which is called "surface" by extension of the familiar surface of a three-dimensional body.
The corresponding surface of an n-dimensional sphere is an n-1 domain with a content proportional with −1 .Using the same method as for the volume element we get The next table shows the volume and the surface of a hyper-sphere for different n.

Table 1 .
The volume and the surface of a n-dimensional sphere