Designing and Building a Cone Frustum

Date: 03/24/2004 at 12:18:06
From: Lynette
Subject: cone frustum question

I am making a pattern for a frustum shape to protect a piece of 
equipment.  I found your information on making a lamp shade pattern 
very helpful.  But since this is large, I am wanting to know how to 
find the distance between the top arcs and bottom arcs in a straight 
line so when I make my compus, I know when to stop and do not have 
to try and measure the arc(s) (about 12') as I go.

I need direction.  Can I put the known information on a plane and 
use a distance formula?  I don't seem to know where to channel my 
thought on getting this information.

Date: 03/24/2004 at 15:28:11
From: Doctor Rick
Subject: Re: cone frustum question

Hi, Lynette.

I believe you are referring to this:

  Pattern for Lampshade
    http://mathforum.org/library/drmath/view/55270.html 

Once you have calculated the radii and the central angle according to 
Dr. Peterson's formulas, you can find the chord lengths and heights 
using formulas found here:

  Segments of Circles
    http://mathforum.org/dr.math/faq/faq.circle.segment.html 

You'll need Case 13: known radius and central angle.  The following 
formulas give the chord length c and the height of the arc, h:

   h = rad(1 - cos(theta/2))
   c = 2*rad*sin(theta/2)

where, from Dr. Peterson's formulas,

             Rs
  rad = S = ---                  for the outer curve
            R-r

               rs
  rad = S-s = ---                for the inner curve
              R-r

  theta = 360(R/S) = 360(R-r)/s  for both curves

The key points of your pattern will be like this:

                   c/2                    c/2
         *----------------------+---------------------*
        /:                      |h                     \
       / :                      *                       \
      /  :                                               \
    s/   :s*cos(theta/2)                                  \s
    /    :                                                 \
   /     :                                                  \
  /      :      C/2                           C/2            \
 *-------+----------------------+-----------------------------*
                                |H
                                *

where I used c and h for the chord and height calculated for the 
inner curve, and C ahd H for the chord and height calculated for the 
outer curve. You'll have to connect the asterisks with arcs to 
complete the pattern.

You may also find the following Dr. Math page useful:

  Formula for Laying Out an Arc
    http://mathforum.org/library/drmath/view/61615.html 

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

Date: 03/28/2004 at 23:59:51
From: lynette
Subject: Thank you (cone frustum question)

Thank you for your quick response!  I was at my wit's end when I found 
the website.  Everything is so well laid out in terms I can 
understand.

I just wanted to say I really appreciated your help. :)