Surface of a Conic Frustum

frus·tum n., pl.frus·tums or frus·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.

Diagram of a conic frustum, aka tapered tube.

I started out realizing that the ratio of the radii R1 and R2 of the funnel was equal to the ratio of the radii Rx and Rx+L of the piece to be cut. From there I simply solved for Rx.

R1 ÷ R2 = Rx ÷ (Rx + L)

R1 (Rx + L) = R2 Rx

R1 Rx + R1 L = R2 Rx

R1 L = R2 Rx - R1 Rx

R1 L = Rx (R2 - R1)

R1 L ÷ (R2 - R1) = Rx

The next question was the angle between the edges of the surface. This is simply the circumference of R1 divided by the circumference of a circle of radius Rx, multiplied by 360.

ø = 360 (2 π R1) ÷ (2 π Rx)

ø = 360 R1 ÷ Rx

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

h = (L2-(R2-R1)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?)

Small end radius
Large end radius
Length
Radius
Angle

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© 2010 W. E. Johns