Lateral Surface of a Cone

Date: 02/10/99 at 21:08:51
From: Grant
Subject: Lateral surface of cone

What is the equation for the lateral surface of a cone?

Date: 02/11/99 at 12:51:55
From: Doctor Peterson
Subject: Re: Lateral surface of cone

Hi, Grant. Welcome to the Doctor's office.

I can take your question two ways: the formula for the lateral surface 
area, or the equation of the surface itself (involving x, y, and z 
coordinates). I'll assume it's the former. You can find the formula on 
this page:

   http://mathforum.org/dr.math/faq/formulas/faq.cone.html   

The area is pi*r*s, where r is the radius of the base, and s is the 
length of the side (from the vertex to the base). You can figure it 
out without too much trouble if you think of the surface as made of a 
piece of paper rolled up. Unroll it flat, and it will be a sector of a 
circle:

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

The arc length of the whole circle would be 2 pi s; but you only have 
as much as will fit around the base of the cone, 2 pi r. That means 
you have

    2 pi r    r
    ------ = ---
    2 pi s    s

of the whole circle, so the area is

     r
    --- * pi s^2 = pi * r * s
     s

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