Solid Angle - Solid Angles in Arbitrary Dimensions

Solid Angles in Arbitrary Dimensions

The solid angle subtended by the full surface of the unit n-sphere (in the geometer's sense) can be defined in any number of dimensions . One often needs this solid angle factor in calculations with spherical symmetry. It is given by the formula

\Omega_{d} = \frac{2\pi^{d/2}}{\Gamma(\frac{d}{2})}

where is the Gamma function. When is an integer, the Gamma function can be computed explicitly. It follows that

\Omega_{d} = \begin{cases} \frac{2\pi^{d/2}}{ \left (\frac{d}{2}-1 \right )!} & d\text{ even} \\ \frac{2^d\left (\frac{d-1}{2} \right )!}{(d-1)!} \pi^{(d-1)/2} & d\text{ odd} \end{cases}

This gives the expected results of 2π rad for the 2D circumference and 4π sr for the 3D sphere. It also throws the slightly less obvious 2 for the 1D case, in which the origin-centered unit "sphere" is the set, which indeed has a measure of 2.

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