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[parent] area of the $n$-sphere (Derivation)

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

$\displaystyle I(n) = \int_{\mathbb{R}^{n+1}} e^{-\sum_{i=1}^{n+1} x_i^2}\, d^{n+1} x. $
Switching to polar coordinates we let $ r^2=\sum_{i=1}^{n+1} x_i^2$ and the integral becomes
$\displaystyle I(n) = \int_{S^n} d\Omega \int_{0}^{\infty} r^{n} e^{-r^2}\, dr. $
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=r^2$, the second integral can be evaluated in terms of the gamma function $ \Gamma(x)$:
$\displaystyle I(n)/A(n) = \frac{1}{2}\int_0^\infty t^{\frac{n-1}{2}} e^{-t}\, dt = \frac{1}{2}\Gamma\left(\frac{n+1}{2}\right). $
We can also evaluate $ I(n)$ directly in Cartesian coordinates:
$\displaystyle I(n) = \left[ \int_{-\infty}^\infty e^{-x^2}\, dx \right]^{n+1} = \pi^{\frac{n+1}{2}}, $
where we have used the standard Gaussian integral $ \int_{-\infty}^\infty e^{-x^2}\, dx = \sqrt{\pi}$.

Finally, we can solve for the area

$\displaystyle A(n) = \frac{2\pi^{\frac{n+1}{2}}}{\Gamma\left(\frac{n+1}{2}\right)}. $
If the radius of the sphere is $ R$ and not $ 1$, the correct area is $ A(n)R^{n}$.

Note that this formula works only for $ n\ge0$. The first few special cases are

$ n=0$
$ \Gamma(1/2)=\sqrt{\pi}$, hence $ A(0)=2$ (in this case, the area just counts the number of points in $ S^0=\{+1,-1\}$);
$ n=1$
$ \Gamma(1)=1$, hence $ A(1)=2\pi$ (this is the familiar result for the circumference of the unit circle);
$ n=2$
$ \Gamma(3/2)=\sqrt{\pi}/2$, hence $ A(2)=4\pi$ (this is the familiar result for the area of the unit sphere);
$ n=3$
$ \Gamma(2)=1$, hence $ A(3)=2\pi^2$;
$ n=4$
$ \Gamma(5/2)=3\sqrt{\pi}/4$, hence $ A(4)=8\pi^2/3$.



"area of the $n$-sphere" is owned by CWoo. [ owner history (1) ]
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See Also: volume of the $n$-sphere, area of a spherical triangle, area of spherical zone


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Cross-references: unit sphere, unit circle, circumference, points, number, sphere, radius, Gaussian integral, Cartesian coordinates, gamma function, terms, variable, polar coordinates, integral, calculate, origin, solid angle, hypersphere, unit, area
There are 3 references to this entry.

This is version 11 of area of the $n$-sphere, born on 2003-07-23, modified 2006-10-18.
Object id is 4495, canonical name is AreaOfTheNSphere.
Accessed 9892 times total.

Classification:
AMS MSC: 51M05 (Geometry :: Real and complex geometry :: Euclidean geometries and generalizations)
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