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Flattened Cone

Date: 08/30/98 at 07:52:58
From: Paul Baker
Subject: Geometric construction of an arc

Dear Dr. Math,

In an earlier geometry class I had, we learned how to take a triangle 
(specificaly the 2d representation of a cone) and, using a compass, 
draw what the cone would look like "rolled out." All I can remember is 
that it had something to do with chord lengths or their relations. 
My teacher didn't know, and my school recently changed textbooks, and 
the new ones for that class no longer include that construction.

Thank you for your time.

Paul Baker

Date: 08/31/98 at 12:16:35
From: Doctor Peterson
Subject: Re: Geometric construction of an arc

Hi, Paul. 

As I understand your question, you want to know how to draw a shape 
that you could cut out and roll up to form a cone whose cross-section 
you are given.

                                   -   *
                                 -      **
                               -          *
                             -            *
                           -               *
                        +-                  *
                        |A                  *
           /\           |                  *
          /  \          |                  *
          /  \          |                  *
         /    \L       L|                **
        /      \        |               *  pi * D
       /        \       |             **
       /        \       |          ***
      /    D     \      |     *****
     --------------     ******

The shape you want to draw will be a sector of a circle with radius 
equal to the slant height (L) of the cone, and arc length equal to the 
circumference (pi * D) of the base. The angle A then will be:

   pi * D                 180 * D
   ------ * 180 degrees = ------- degrees
   pi * L                    L

If you are looking for a compass-and-straightedge construction of this 
angle, my first impression is that it may be impossible.

Let me know if this is not what you wanted.

- Doctor Peterson, The Math Forum
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Associated Topics:
High School Conic Sections/Circles
High School Geometry
High School Higher-Dimensional Geometry

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