Construction of Cone Surface

Date: 10/12/2002 at 11:00:19
From: Albert 
Subject: Construction of cone surface

What is the relation between the angle of a cone at the vertex and the 
flat angle (theta) of the developed surface on a plane?

Example: I want to build a cone with a 30-degree aperture angle at the 
vertex, height h, and base radius r. What is the angle at the vertex 
of the circle sector when I develop that surface on the floor prior to 
cutting the material?  

Is there a general formula or function theta = f(alpha)? 
Alpha = angle at the vertex.

Thank you.

Date: 10/12/2002 at 23:22:10
From: Doctor Peterson
Subject: Re: Construction of cone surface

Hi, Albert.

This is a common question; I searched our site for "cone surface 
sector" and found this, among others, which gives a picture:

   Lateral Surface of a Cone
   http://mathforum.org/library/drmath/view/55082.html 

Let's copy that picture and add a side view of your cone:

                      *********
                ******         ******
             ***                   /
           **                     /
         **                      /
        *                       /
       *                       /s
      *                       /
     *                       /
     *                      /
    *                      /
    *                     +---------------------*
    *                theta                      *
     *                                         *
     *                                         *
      *                                       *
       *                                     *
        *                                   *
         **                               **
           **                           **
             ***                     *** 2 pi r
                ******         ******
                      *********

                          +
                         /|\
                        / | \
                       /alpha\
                      /   |   \
                     /    |    \s
                    /     |h    \
                   /      |      \
                  /       |       \
                 /        |        \
                +---------+---------+
                     r         r

The angle theta can be found by taking the ratio of the arc length, 
2 pi r, to the circumference of the whole circle, 2 pi s, and 
multiplying 360 degrees by that ratio. This is the angle r/s times 
360 degrees. Since your angle alpha satisfies

    sin(alpha/2) = r/s

you just have to multiply 360 degrees by this sine.

If you have any further questions, feel free to write back.

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