Volume of Unit Hypersphere

Theorem

The volume of the unit sphere in $n$-dimensional space increases as $n$ goes up to $5$, but decreases thereafter.

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Sequence of Volumes of Unit Hyperspheres

The sequence of volumes of the unit sphere in $n$-dimensional space begins as follows:

$\ds n = 1$

$:$

$\ds \map V 1 = 2$

$\ds n = 2$

$:$

$\ds \map V 2 = 3.1$

$\ds n = 3$

$:$

$\ds \map V 3 = 4.2$

$\ds n = 4$

$:$

$\ds \map V 4 = 4.9$

$\ds n = 5$

$:$

$\ds \map V 5 = 5.264$

$\ds n = 6$

$:$

$\ds \map V 6 = 5.2$

$\ds n = 7$

$:$

$\ds \map V 7 = 4.7$

Also see

Maximum Volume of Unit Radius Sphere in Fractional Dimensions

Sources

1986: David Wells: Curious and Interesting Numbers ... (previous) ... (next): $5$
1997: David Wells: Curious and Interesting Numbers (2nd ed.) ... (previous) ... (next): $5$