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

### Determining Cone's Original Dimensions from a Slice

Date: 05/18/99 at 11:33:33
From: Mike Branson
Subject: Truncated cones revisited

Dr. Math,

You have helped us tremendously with the truncated cone problem I
presented you several weeks ago. Thank you.

I have another question for you along the same lines. We have a
customer who has provided us with some dimensions, but does not know
how to measure the angle. We have a large arc length of 64", a small
arc length of 41", and a length of 26" between these two arc lengths.
Do you know how we can make this work?

Thanks again,

Mike Branson of Sefar America

Date: 05/18/99 at 16:00:36
From: Doctor Rob
Subject: Re: Truncated cones revisited

Thanks for writing to Ask Dr. Math!

I think the situation you are describing is as follows:

41
__,,---''''---..__
_,--'                  `--._
_,-'                            `-._
o--------------------------------------o
`.                26                ,'
`._                            _,'
`._                      _,'
`-._              _,-'
``--..__,,--''
64

You can use the formulas on the following Ask Dr. Math FAQ web page to
find out what you need to know about these two segments with a common
chord:

Circle Formulas: Segment of a Circle
http://mathforum.org/dr.math/faq/formulas/faq.circle.html#segment

If this is not what you meant, write again.

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

Date: 05/18/99 at 16:53:33
From: Mike Branson
Subject: Re: Truncated cones revisited

Dr. Math,

I think I might need to explain my scenario a little better. The
dimensions I gave you could be most easily visualized as a truncated
cone which is cut down the seam and laid flat. We can draw it on
autocad by first drawing a circle and offsetting it with another
circle by 26". Then we draw a 90 degree angle from the center and
remove the portion of the two circles outside of the ninety degree
angle. Next we needed to play with the angle to allow us to have one
arc of 64" and one arc of 41". It looks sort of like this:

+  +   64
+                  +
+          +  +            +
+         +      41    +           +
+        +                     +         +
+------+                            +------+
<- 26 ->                            <- 26 ->

My question is: Given the arc length, can you determine the
circumference of a circle? And, if given the above figure, can you
calculate the angle (or an angle) that is needed?

Date: 05/18/99 at 17:04:54
From: Doctor Peterson
Subject: Re: Truncated cones revisited

Hi, Mike. I see Doctor Rob got to you before me, without the knowledge

Here's my understanding of your problem:

***********
******...........******  S2=64"
*** \.....................***
***     \.......................***
*         \.........................*
*      L=26"\.........................*
**             \.........................**
*                \..........................*
*                  \.*****.S1=41".............*
*                 ***     ***.................*
*                 *   \       *.................*
*                *     \A      *................*
*                *      +------*----------------*
*                *      |  R1  *     L=26"      *
*                 *     |     *                 *
*                 ***  |  ***                 *
*                    **|**                    *
*                     |                     *
**                   |R2                 **
*                  |                  *
**                |                **
**              |              **
***           |           ***
*******    |    *******
*********

Given two arc lengths S1 and S2, and a segment length L, you need to
find the two radii R1 and R2 and the angle A. Three numbers to find
three numbers - sounds feasible!

Let's write equations for the numbers you have in terms of the numbers
we want:

S1 = A * R1    (A is the angle in radians)
S2 = A * R2
L = R2 - R1

Now we can solve this for A, R1, and R2:

S2 - S1 = A(R2 - R1) = A*L

so
S2 - S1
L

S1   S1 * L
R1 = -- = -------
A    S2 - S1

S2   S2 * L
R2 = -- = -------
A    S2 - S1

180 deg
A = (64 - 41)/26 =  0.8846 radians * ---------- = 50.68 degrees
41*26
R1 = ----- = 46.35 inches
64-41

64*26
R2 = ----- = 72.35 inches
64-41

As before, of course, this doesn't include any allowances for seams. I
think the only real difference from the last time is that you have
given the circumference rather than the diameter of the end circles of
your cone. The formulas this time should be a little easier to follow.

Incidentally, if you are interested in the vertex angle of the cone
itself (after it is rolled up), that will be the arcsin of the ratio
of
the base radius to the slant height, or

S2               A              S2-S1
A_vertex = arcsin(-------) = arcsin(----) = arcsin(------)
2 pi R2           2 pi           2 pi L

= arcsin(0.8846/6.28) = 8.09 degrees

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

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