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Diameter of Cone Frustum


Date: 05/24/2000 at 05:39:25
From: Vin
Subject: Diameter of cone

Dear Dr. Math,

I need to solve for the diameter of a cone when it is opened up flat. 
The inner and outer diameters, and the height are given. The top part 
of the cone is cut of by the inner diameter. 

Kindly help me.


Date: 05/24/2000 at 09:05:42
From: Doctor Peterson
Subject: Re: Diameter of cone

Hi, Vin.

It's not quite clear whether the inner and outer diameters you are 
given refer to the bases of the frustum of a cone, or to the flattened 
shape; I'll assume you mean you are given D, d, and h in this side 
view of a frustum:

                  d
             +----+----+
            /     |     \
           /      |      \
          /       |       \
         /        |h       \
        /         |         \
       /          |          \
      /           |           \
     +------------+------------+
                  D

You can find the total height of the uncut cone using similar 
triangles:

                  + -----------------
                 /|\              |
                / | \l            |
               /  |  \            |
              /   |  r\           |
             +----+----+          |
           L/     |     \         H
           /      |      \        |
          /       |       \       |
         /        |h       \      |
        /         |         \     |
       /          |          \    |
      /           |           \   |
     +------------+------------+ ----
                R

Then the radii of the circles you get when you flatten the cone are 
the slant heights L and l of the cone, which you can find using the 
Pythagorean theorem.

Write back if you need more help.

- Doctor Peterson, The Math Forum
  http://mathforum.org/dr.math/   
    
Associated Topics:
High School Geometry
High School Higher-Dimensional Geometry

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