volume of the n-sphere

The volume contained inside Sn, the n-sphere (or hypersphereMathworldPlanetmath), is given by the integral


Going to polar coordinates (r2=i=1n+1xi2) this becomes


The first integral is the integral over all solid angles subtended by the sphere and is equal to its area A(n)=2πn+12Γ(n+12), where Γ(x) is the gamma functionDlmfDlmfMathworldPlanetmath. The second integral is elementary and evaluates to 01rn𝑑r=1/(n+1).

Finally, the volume is


If the sphere has radius R instead of 1, then the correct volume is V(n)Rn+1.

Note that this formulaMathworldPlanetmathPlanetmath works for n0. The first few cases are

  • n=0

    Γ(3/2)=π/2, hence V(0)=2 (this is the length of the intervalMathworldPlanetmath [-1,1] in );

  • n=1

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

  • n=2

    Γ(5/2)=3π/4, hence V(2)=4π/3 (this is the familiar result for the volume of the sphere);

  • n=3

    Γ(3)=2, hence V(3)=π2/2.

Title volume of the n-sphere
Canonical name VolumeOfTheNsphere
Date of creation 2013-03-22 13:47:09
Last modified on 2013-03-22 13:47:09
Owner CWoo (3771)
Last modified by CWoo (3771)
Numerical id 10
Author CWoo (3771)
Entry type Derivation
Classification msc 51M05
Related topic AreaOfTheNSphere