Associated Topics || Dr. Math Home || Search Dr. Math

### 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.

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/
```
Associated Topics:
High School Geometry
High School Higher-Dimensional Geometry

Search the Dr. Math Library:

 Find items containing (put spaces between keywords):   Click only once for faster results: [ Choose "whole words" when searching for a word like age.] all keywords, in any order at least one, that exact phrase parts of words whole words

Submit your own question to Dr. Math
Math Forum Home || Math Library || Quick Reference || Math Forum Search