Pappus' Centroid Theorem

Date: 06/05/2003 at 21:43:24
From: Tyler
Subject: What is the formula for the volume of a doughnut-shaped 
cylinder?

I am in an eighth grade pre-algebra math class. We are learning 
abought cylinders and prisms. I asked my teacher for the formula for 
the volume of a cylinder with unequal sides, such as a perfect 
doughnut. He didn't know, but I came up with a possible formula and I 
would like to check it.

My Formula: 

Volume = the circumference of the outer circle + 
         the circumference of the inner circle (hole in the middle) 
         divided by two (this part is to = out the lengths of the 
         sides) 
         times the area of the thickness of the doughnut

   The diameter across the whole doughnut is d1.
   The diameter across the hole in the doughnut is d2.
   The diameter across the width of the doughnut is d3.

   (pi times d1 + pi times d2 divided by 2) times
   (pi (d3 devided by 2) to the 2nd power = Volume

Date: 06/06/2003 at 09:17:04
From: Doctor Peterson
Subject: Re: What is the formula for the volume of a doughnut-shaped 
cylinder?

Hi, Tyler.

It happens that your formula is correct; alternatively, you can just 
take the circumference of the circle made by the CENTER of the small 
circular cross-section, and multiply that by the area of that cross-
section.

This fact is a consequence of the centroid theorem of Pappus, which I 
discussed here:

  Volume of a Cone
  http://mathforum.org/library/drmath/view/51840.html 

It's not obvious at first glance that your formula HAS to be true, but 
it does make sense that at least some sort of average should work; 
Pappus shows just what sort of average it actually takes to make it 
work.

You can find the formula for volume of a circular torus, which is 
what you are asking about, in the Dr. Math Geometric Formulas FAQ:

  Ellipsoid, Torus, Spherical Polygon Formulas
  http://mathforum.org/dr.math/faq/formulas/faq.ellipsoid.html#torus 

  V = 2 pi^2 R r^2

which, using the approach I gave above, means

  V = C * A = (2 pi R)(pi r^2)

Note that, though the picture doesn't make it clear, R is the radius 
from the center of the whole torus to the center of the small 
circular cross-section; and r is the radius of that small circle.

          *****=======================*****
      ****  |  ****               ****  |  ****
     *      |      *             *      |      *
    *       |       *           *       |       *
    *       +-------*-----+-----*-------+       *   front view
    *               *     |     *       |       *
     *             *      |      *      |      *|
      ****     ****       |       ****  |  **** |
          *****===========|===========*****     |
                          |             |       |
                          |<-----R----->|<--r-->|
                          |             |       |

                      *********
                ******         ******
             ***                     ***
           **                           **
         **           ...........         **
        *          ...           ...        *
       *        ...                 ...      *
      *        .         *****         .      *
     *        .       ***     ***       .      *
     *       .       *           *       .     *
    *       .       *             *       .     *
    *       .       *      <------*------>.<--->*    top view
    *       .       *             *R      .  r  *
     *       .       *           *       .     *
     *        .       ***     ***       .      *
      *        .         *****         .      *
       *        ...                 ...      *
        *          ...           ...        *
         **           ...........         **
           **                           **
             ***                     ***
                ******         ******
                      *********

You've done some good thinking!

If you have any further questions, feel free to write back.

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