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.

Proof

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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): $5$