Pattern for Lampshade

Date: 02/17/2001 at 16:41:23
From: Loraine Green
Subject: Formula for making pattern for lamp shade

I would like to know the formula for computing the surface area of a 
lampshade so that I can make a pattern from it. I know there is a 
circle at the top, a bigger circle at the bottom, and a given 
distance between top and bottom, but I don't know how to use this 
information.  Your help will be greatly appreciated.

Loraine Green

Date: 02/17/2001 at 22:56:54
From: Doctor Peterson
Subject: Re: Formula for making pattern for lamp shade

Hi, Loraine.

This is a fairly common question; many things have this shape, called 
the frustum of a cone; but a search of our archives didn't reveal an 
answer that gives the actual formulas for laying out the pattern. The 
closest I could come was this, which shows how to find the surface 
area, but doesn't directly tell about the pattern:

  Volume and Surface Area of a Cone Frustum
  http://mathforum.org/dr.math/problems/reddy.8.30.99.html   

See the picture there, and here's a formula that will be useful:

                 Rs
             S = ---
                 R-r

This says how to find the slant height S of the complete cone, which 
is the radius of the circle from which the frustum's lateral surface 
will be cut, from the slant height s of the frustum, and the radii r 
and R of the top and bottom circles. The radius of the smaller circle 
you'll have to cut is S-s, and the angle at the center of these 
circles will be the ratio of the circumference of the base of the 
frustum to the circumference of the circle with radius S, times 360 
degrees:

             A = 360(R/S) = 360(R-r)/s

So make this pattern, using the indicated measurements:

                        ***********
                    ******...........****** 2 pi R
                 *** \.....................***
              ***     \.......................***
             *         \.........................*
            *           \.........................*
          **             \.........................**
         *                \..........................*
        *                  \.*****.2 pi r.............*
        *                 ***     ***.................*
       *                 *   \       *.................*
       *                *     \ A     *................*
       *                *      +------*----------------*
       *                *      | S-s  *        s       *
       *                 *     |     *                 *
        *                 ***  |  ***                 *
        *                    **|**                    *
         *                     |                     *
          **                   |S                  **
            *                  |                  *
             **                |                **
               **              |              **
                 ***           |           ***
                    *******    |    *******
                           *********

That is, using the same center make circles with radii S-s and S; then 
mark out angle A at the center and use the shaded part to form the 
lampshade.

If you know h rather than s, you can find s using

    s = sqrt(h^2 + (R-r)^2)

Write back if you need more help using these formulas.

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