I have been intrigued with this shortcut, which you may find of interest. Cones have several uses, such as sheet-metal chimney caps, but the layout of a cone is puzzling to some craftsmen. Here is a fast method that doesn’t require higher mathematics. The example shows how to lay out a cone 6 in. high with a 24-in. dia. base.
Using the slant length as radius, scribe a circle on the material, and snip. Then measure the base circumference of the cone on the circle, adding at least a 3/4-in. overlap for screws, and cut out the wedge.
—William A. Julien, S. Chatham, MA
Edited and illustrated by Charles Miller
From Fine Homebuilding #4
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The technique still doesn't explain how to measure that circumference of 75 3/8 inches. One can only presume that one cuts out the circle and then tries to wrap a tape measure around it, marking off the correct length. ....And the definition of frustration is...
With one simple additional formula one can measure the exact number of degrees required for that circle. Everyone has an old drafting protractor (and squares, compass and dividers) in their workshop....
No. of degrees = (360 x 75 3/8) / (2 x pi x 13 7/16)
= 321.6 degrees + 3/4" overlap
(Yes, my protractor is graduated in 1/2 degrees, so that 4th significant digit does have meaning.)
For clarity I've shown the formula using fractions like the author. In reality it is far easier to perform this calculation using decimals as one would have originally obtained using a calculator (or slide rule) ... and yes, I still have a slide rule in my workshop from days gone by.