Hans Walser, 

Sphere and cylinder

# 1     The original problem

In (Richeson 2017, p 23) I found:

“Just noticed that the ratio of the volume of a sphere to the volume of the cylinder containing it is 2:3.

Likewise, the surface area. Eureka!” Fir. 1: Sphere and cylinder

# 2     In other dimensions

The mentioned statement holds similar in other dimensions.

We use the following notations: = Volume of the sphere in the n-dimensional space = Surface of the sphere in the n-dimensional space = Volume of the cylinder in the n-dimensional space = Surface of the cylinder in the n-dimensional space

Examples:   Ratio  Ratio 2      3      4      5      6      7      Tab. 1: Examples

In every dimension there is the same ratio. In even dimensions the ratio is irrational.

# 3     General case

Notation: (1)

Hence we get: (2)

And for the volume of the cylinder: (3)

For the surface of the cylinder we get: (4)

Remark: (5)

The volumes of the sphere and the cylinder have the ratio: (6)

For the surfaces we get the ratio: (7)

From (6) and (7) we see that the ratios are equal in any dimension.

# 4     Explicit formulas

According to  we have: (8)

From (6) and (8) we get:

In even dimensions we have:

ratio = (9)

!! denotes the double faktorial, defined for odd integers by: (10)

In odd dimensions we get:

ratio = (11)

References

Richeson, David (2017): A-Tweeting We Will Go. Building a Professional Network with Twitter. MAA FOCUS | JUNE/JULY 2017 | maa.org/focus. 22-25.

Websites

 Wikipedia: n-sphere

https://en.wikipedia.org/wiki/N-sphere#Volume_and_surface_area