Conical Sections

Date: 08/02/99 at 11:00:48
From: Dave Cope
Subject: Conical sections

If I know the angle and the diameter of one end of a cone and the 
length of the cone, what is the diameter of the other end of the cone?

Date: 08/02/99 at 13:15:38
From: Doctor Peterson
Subject: Re: Conical sections

Hi, Dave.

Since a cone has only one end, I'll assume you're referring to a 
frustum of a cone, like a cylinder with sloped sides. I'm also not 
sure what angle you're referring to, so I'll take it to be the angle 
between the sides and vertical. Also I'll assume you know the actual 
(perpendicular) height, not the slant height along the side. Then you 
have this:

                    +
                   /|\
                  / |A\
                 /  |  \
                /   |   \
               /    | r1 \
              +-----+-----+
             /      |     |\
            /       |     |A\
           /        |     |  \
          /         |     |   \
         /          |     |    \
        /          h|     |     \
       /            |     |      \
      /             |     |       \
     /              |     |        \
    /               |     |  r2-r1  \
   +----------------+-----+----------+
                            r2

Then we have a right triangle with legs h and r2-r1 and angle A; 
trigonometry gives us

     r2 - r1
     ------- = tan A
        h

so

     r2 = r1 + h tan A

If any of these measurements are not what you have, or if you need 
help finding the tangent of the angle, let me know.

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