area of the n-sphere

The area of Sn the unit n-sphere (or hypersphereMathworldPlanetmath) is the same as the total solid angle it subtends at the origin. To calculate it, consider the following integral


Switching to polar coordinatesMathworldPlanetmath we let r2=i=1n+1xi2 and the integral becomes


The first integral is the integral over all solid angles and is exactly what we want to evaluate. Let us denote it by A(n). With the change of variable t=r2, the second integral can be evaluated in terms of the gamma functionDlmfDlmfMathworldPlanetmath Γ(x):


We can also evaluate I(n) directly in Cartesian coordinatesMathworldPlanetmath:


where we have used the standard Gaussian integral -e-x2𝑑x=π.

Finally, we can solve for the area


If the radius of the sphere is R and not 1, the correct area is A(n)Rn.

Note that this formula works only for n0. The first few special cases are

  • n=0

    Γ(1/2)=π, hence A(0)=2 (in this case, the area just counts the number of points in S0={+1,-1});

  • n=1

    Γ(1)=1, hence A(1)=2π (this is the familiar result for the circumferenceMathworldPlanetmath of the unit circle);

  • n=2

    Γ(3/2)=π/2, hence A(2)=4π (this is the familiar result for the area of the unit sphere);

  • n=3

    Γ(2)=1, hence A(3)=2π2;

  • n=4

    Γ(5/2)=3π/4, hence A(4)=8π2/3.

Title area of the n-sphere
Canonical name AreaOfTheNsphere
Date of creation 2013-03-22 13:47:06
Last modified on 2013-03-22 13:47:06
Owner CWoo (3771)
Last modified by CWoo (3771)
Numerical id 14
Author CWoo (3771)
Entry type Derivation
Classification msc 51M05
Related topic VolumeOfTheNSphere
Related topic AreaOfASphericalTriangle
Related topic AreaOfSphericalZone