Cylinder Height, Area, Volume

Date: 04/07/2003 at 12:22:21
From: James Boulter
Subject: Cylinder Height

Dear Sirs,

I have recently been working on an investigation into cylinders and 
how their volume is affected by their dimensions. The cylinder is 
open-ended and is formed by rolling up a piece of card whose height 
is x and with a fixed area A. The formula for the volume given the 
area of card and height of card is A^2/(4*PI*x). Supposing the area 
is 1200cm^2 and the height 20cm, the volume would therefore be 5730 
cm^3.

What is confusing is that if you keep on making the height smaller 
and smaller (i.e. the radius of the top becomes larger and larger), 
the volume keeps on going up and up. I have tried the formula for such 
low height values as 10^-6, and the volume gets higher and higher.  
Surely this defies physics? If the height is as small as physically 
possible (i.e. one atom thick), the volume would be huge. But I don't 
understand this because the cylinder wouldn't actually be able to hold 
anything. Perhaps you could explain why the volume gets larger and 
larger and tell me where I may be going wrong.

Date: 04/07/2003 at 23:38:33
From: Doctor Ian
Subject: Re: Cylinder Height

Hi James, 

I suppose the place to start might be by examining the formula you're
using.  If the card has height x, and area A, then its width is

   w = A/x

But the width of the card is the circumference of the cylinder. If the 
radius of the cylinder is r, then

           w = 2 pi r

         A/x = 2 pi r

  A/(2 pi x) = r

The volume of the cylinder will be 

  volume = pi r^2 x

         = pi [A/(2 pi x)]^2 x

           pi A^2 x
         = ----------
           4 pi^2 x^2

             A^2 
         = ------
           4 pi x

Which is what you have. Good!

Now look at the equation and think about what it _means_. Since A is 
fixed, the only thing we can vary is x, the height of the cylinder. 
And where is x? In the denominator. So if we make x greater, the value 
of the fraction - that is, the volume - has to get smaller.  

Conversely, if we make x smaller, the value of the fraction has to get
bigger. So mathematically, this is what's going on. 

What about physically?  

Note that we often think of the 'volume' of something as the amount of
stuff it can hold, but it's really a measure of how big something else
needs to be to hold _it_ (i.e., how much space it takes up). After 
all, a solid lump of metal can't hold anything, but we would say that 
it has a volume, right? So just because something is too flat to hold 
anything doesn't mean it can't have a large volume.  

(If it helps, imagine buying a ream of paper. This definitely has a
volume. Now spread the pieces out, one deep, over a gym floor. Now you 
have a lot of area, and hardly any height. But the volume of the paper 
is exactly the same.)

Suppose we construct a particular cylinder with height x, and then 
slice it in half horizontally, so that we have two cylinders with
height x/2.  Now we put the two cylinders next to each other. 

So far, the volume of the two cylinders is the same as the volume of
the cylinder that we started with, right? We have half the height, but 
twice the area, so it evens out. 

But now let's make a vertical cut in the side of the smaller 
cylinders, so we can roll them out and combine their circumferences to
make one big circle. What will the area of the larger circle be?

Well, if we start with two circles of radius r, the total area is

  area = 2 * (pi r^2)

The circumference of each circle is 

  circumference = 2 pi r

So when we combine them, the circumference of the larger circle is 

  circumference' = 2 * (2 pi r)

                 = 4 pi r

What is the area of this circle?  It's 

  area' = pi (2r)^2

        = 4 pi r^2

which is twice the area we started with. So when we combine two 
circles in this way, we get a larger area. (You might have fun
thinking about why this should be the case.) 

So here's a way to think about what's happening with the cylinder, as
we shrink the height. We can shrink it by cutting the cylinder in two, 
then combining the circumferences of the two pieces. Each time we do 
this, we cut the height by a factor of two, but we increase the _area_ 
by more than a factor of two. 

That is, every time we shrink the cylinder, we more than gain in area
what we give up in height.  So the more you shrink the height, the
more you increase the volume. 

Does this make sense? 

- Doctor Ian, The Math Forum
  http://mathforum.org/dr.math/