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 
of your earlier question.

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
    A = ------- radians
           L

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

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

With your numbers,
                                         180 deg
    A = (64 - 41)/26 =  0.8846 radians * ---------- = 50.68 degrees
                                         pi radians
         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/