Volume of a Truncated Cylinder

Date: 05/07/2000 at 11:40:52
From: Dexter Evans
Subject: Formula for the volume of a truncated cylinder

Hi,

I'm looking for the formula to find the volume of a cylinder, but not 
just any cylinder; this cylinder has had part of its top removed. If 
the cut had been at a right angle to the height of the cylinder, then 
it would simply be a shorter cylinder. The problem is that the cut was 
at an angle.To make things more difficult, the cut was initiated at a 
certain distance below the top.  Help!

Thanks,
Garen Evans

Date: 05/07/2000 at 15:18:21
From: Doctor Schwa
Subject: Re: Formula for the volume of a truncated cylinder

You can probably take two copies of your cylinder, one of them turned 
upside-down, and put them together to make one perfect cylinder. Can 
you see what I'm talking about? If not, please write us back.

Then the volume of your cylinder is just half that of the big 
cylinder.

You might also find the Dr. Math FAQ useful:

   http://mathforum.org/dr.math/faq/   

Look in the section on "Formulas" from the link at the top of the 
page.  

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

Date: 05/08/2000 at 11:58:34
From: Garen Evans
Subject: Re: Formula for the volume of a truncated cylinder

Hi Dr. Math,

Well that's a good idea, thanks. Unfortunately it's a bit more 
complicated. You see the cylinder is cut at different angles at the 
top and at the bottom.  Also I'd like to be able to find the volume of 
the chunk(s) that I removed, which of course should follow quite 
naturally if you knew the remaining volume. I'm stumped and I can't 
find any references to this type of problem. Got any more ideas?

Thanks,
Dexter and Garen

Date: 05/08/2000 at 13:43:16
From: Doctor Schwa
Subject: Re: Formula for the volume of a truncated cylinder

Okay then - you're going to need to describe the shape of the cuts in 
more detail. Is it two flat planar cuts, one at one angle and a second 
one at another angle? Do they meet in the middle at a nice sharp line, 
or are the cuts not flat, so there's a continuous curve in between?

This problem could get pretty hard!

If you have (or can make) a formula that gives the height of the 
cylinder as a function of the (x,y) coordinate of the base (that is, 
how tall is the column sticking up from a given spot?), then that 
would give one way to find the volume.

In any case, there's no way to find the volume without knowing a 
little bit more about the exact shape.

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

Date: 05/08/2000 at 15:18:26
From: Garen Evans
Subject: Re: Formula for the volume of a truncated cylinder

Okay great! More information: I'm growing fond of my cylinder so let's 
call it Nancy. I'm going to set Nancy the Cylinder upright. Nancy is a 
right cylinder with the same diameter from one end to the other. The 
diameter is 4 inches. The length of the cylinder can be measured from 
A to C or from B to D (same length both sides, obviously). Nancy wants 
a haircut, so let's take off BJ. Then she wants one of her high heels 
brought down a notch, so let's also cut along QR. All cuts are 
perpendicular to the drawing below. So to answer your question, 
they're flat, planar type cuts. I guess it'd be interesting if they 
were curved cuts, but it's not necessary.

     AB = 4 inches
     AC = BD = 17 feet
     AJ = 3 feet
     CQ = 3 feet
     DR = 6 feet

Here's the side drawing:

           TOP
     A +----------+ B
       |          |
       |          |
     J *          |
       |          |
       |          |
       |          |
       |          |
       |          |
       |          * R
       |          |
       |          |
     Q *          |
       |          |
       |          |
     C +----------+ D
          BOTTOM

As you can see, one cut is from the edge of the cylinder, while the 
cut at the bottom starts a way up from the bottom of the cylinder.

Garen and Dexter

Date: 05/10/2000 at 13:15:35
From: Doctor Schwa
Subject: Re: Formula for the volume of a truncated cylinder

Okay, so you can find the volume of ABCD easily, right? And you can 
use the method we talked about last time (imagining it joined with a 
flipped-over copy of itself) to find the volume of CDRQ and ABJ. Then 
subtract!

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