frus·tumn.,pl.frus·tumsorfrus·ta.Mathematics.The part of a solid, such as a cone or pyramid, between two parallel planes cutting the solid, especially the section between the base and a plane parallel to the base. [Latin, piece broken off.]

When in the Course of human events, it becomes necessary for a tinkerer to build a conic frustum, also known as a tapered tube or a funnel, a little math goes a long way.

I started out realizing that the ratio of the radii R_{1} and R_{2} of the funnel was equal to the ratio of the radii R_{x} and R_{x}+L of the piece to be cut. From there I simply solved for R_{x}.

**R _{1} ÷ R_{2} = R_{x} ÷ (R_{x} + L)**

**R _{1} (R_{x} + L) = R_{2} R_{x}**

**R _{1} R_{x} + R_{1} L = R_{2} R_{x}**

**R _{1} L = R_{2} R_{x} - R_{1} R_{x}**

**R _{1} L = R_{x} (R_{2} - R_{1})**

**R _{1} L ÷ (R_{2} - R_{1}) = R_{x}**

The next question was the angle between the edges of the surface. This is simply the circumference of R_{1} divided by the circumference of a circle of radius R_{x}, multiplied by 360.

**ø = 360 (2 π R _{1}) ÷ (2 π R_{x})**

**ø = 360 R _{1} ÷ R_{x}**

Or, if you start with the layout and want to develop the frustrum:

**h = (L ^{2}-(R_{2}-R_{1})^{2})^{½}**

And that's all it takes. All you need to do is remember to add a tab on one edge so you can overlap the ends. Here is a calculator to help you avoid some FRUSTRAtion. (Get it?)

© 2010 W. E. Johns