Diameter of the Base of a Cone

Date: 08/12/98 at 10:58:25
From: ALAN BROWN
Subject: Diameter of the Base of a Cone

I am having difficulty remembering and/or finding the formula that 
would allow me to calculate the diameter of the base of a cone of nine 
degrees at various lengths. Could you help?

The question would be: how many feet wide is the "footprint" of a cone 
of nine degrees at various depths or distances?

Thanks,
Alan Brown

Date: 08/12/98 at 14:02:07
From: Doctor Mike
Subject: Re: Diameter of the Base of a Cone

Hi Alan,   

This is an interesting question, because you can also ask it without 
using either the word cone or the word diameter! If you look at the 
cone from the front (like an ice-cream cone standing upside-down on a 
table, and your eye is about an inch above the plane of the tabletop) 
it looks like an isosceles triangle. Isosceles means 2 equal sides. 
               
          /\
         /18\
        /    \
       /      \       H feet high
      /        \
     /          \
    /            \
   /______________\
    
I am guessing that 18 degrees is the entire angle at the top. Usually 
in discussions about cones we use the "half-angle" which is the angle 
between the sloping side and the vertical. If the full angle is 9, the 
half-angle would of course be 4.5. Let's split the isosceles triangle 
down the middle to get a right triangle.     
                
     |\
     |9\     
     |  \
     |   \        
   H |    \
     |     \
     |      \
     |_______\  
         R  
   
Now we need to introduce some trigonometry into our problems. The 
tangent of 9 degrees is Opposite over Adjacent, or in this case R/H.  
Because you know H and want to know R, the most useful version of the 
formula is:
   
   R = H * tan(9)  
   
Actually, that would correspond to the Radius of the base of the cone,
and what you really want is twice that for the diameter, or:
   
   D =  2 * H * tan(9)   
   
What's that? You don't know the tangent of 9 degrees? Whip out your
calculator or use 0.15838444, which should be close enough. If you need 
tan(4.5), that is 0.078701707, but go ahead and buy yourself a simple 
scientific calculator. You can probably get one for under $10.
   
I hope this helps.    
  
- Doctor Mike, The Math Forum
Check out our web site! http://mathforum.org/dr.math/