Pattern for LampshadeDate: 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/ |
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