Volume of a Frustum Cone

Date: 03/20/2001 at 12:07:21
From: Randy Zerr
Subject: Volume of a Frustum Cone

I have a frustum cone with the bottom radius being 4 inches and the 
top radius being 2 inches with a vertical height of 12 inches.  I am 
trying to figure out what the interior heights would be if I split 
the volume into equal thirds.  I know that the bottom height would be 
less than the middle height because there is more volume in the 
bottom of the frustum cone.  I also know that the middle height 
should be more than the bottom height but less than the top height, 
because the cone gets smaller as I move upward, but the volume is the 
same for the middle as for the bottom and the top. I have tried to 
draw it out below for you to see what I am looking for.

                4" dia.
               ----  ----------------
              /    \               |
             /  H3  \              |
            /        \             | 
           /----------\            |
          /            \          12"
         /      H2      \          |
        /                \         |
       /------------------\        |
      /         H1         \       |
     /                      \      |
     ------------------------  ------           
              8" dia.

Can you please help me?  I have been working on this problem for two 
days, and I keep coming up with different answers.

Date: 03/20/2001 at 13:02:46
From: Doctor Peterson
Subject: Re: Volume of a Frustum Cone

Hi, Randy.

Here's one way to approach it. Extend your frustum to a complete cone, 
whose height will be 24":

                         +
                        /|\
                       / | \
                      /  |  \
                     /   |   \
                    /  12|    \
                   /     |     \
                  /      |      \
                 /       |       \
     A          +--------+--------+
               /        x|         \
              /          |          \
     B       +-----------+-----------+
            /           y|            \
           /             |             \
     C    +--------------+--------------+
         /              z|               \
        /                |                \
     D +-----------------+-----------------+
                                  r

You want to make a sequence of four cones: the whole cone from which 
the frustum was made (whose base is shown as D), the part cut off the 
top (A), and two others (B, C) in between. Their volumes have to 
differ by a common amount, which will be 1/3 the volume of the 
frustum. They will all be similar (the same angle); so their volumes 
will be proportional to the cube of their heights. The heights will be

    12
    12 + x
    12 + x + y
    12 + x + y + z = 24

so their volumes will be proportional to 12^3, (12+x)^3, (12+x+y)^3, 
and 24^3.

See if you can determine, first, what each volume must be in order to 
make the three differences equal; then, what each height must be; and 
then what x, y, and z must be.

(You may have noticed that I don't care about the radius at the bottom 
- it doesn't matter.)

If you need more help, write back and show me what you have done.

Another method you may want to try is to look up the formula for the 
volume of a frustum in our FAQ (under "Formulas"), and write an 
equation saying that the volume of the bottom frustum (with height h) 
is 1/3 that of the whole. You'll have to use similar triangles to 
determine the upper radius of this smaller frustum.

  Geometric Formulas - Dr. Math FAQ
  http://mathforum.org/dr.math/faq/formulas/   

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