# Volume of Unit Hypersphere

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## 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

## Sources

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