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### 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
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:

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:

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. :)
```
Associated Topics:
High School Conic Sections/Circles
High School Geometry
High School Higher-Dimensional Geometry
High School Practical Geometry

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