Hans Walser, [20170711]

Sphere and cylinder

# 1     The original problem

In (Richeson 2017, p 23) I found:

ŇJust 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!Ó

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 [1] 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

[1] Wikipedia: n-sphere

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