Thursday, January 26, 2012

Making a Cone from a Flat Sheet

I wanted to make a transition fitting for my dust collector, that would go from 10" diameter at the top to 4" diameter at the bottom - basically a big funnel to drop sawdust into a bucket.  I remembered from pre-school that you can take a circle of paper, cut a wedge out of it, and it will make a cone.  Cut the bottom off the cone, and there you go.  Yeah, not so easy if you actually want to make a shape with predefined dimensions.  After a few days of head scratching, I figured out some fairly simple formulas to cut a cone of whatever size I want out of paper, plastic, or sheet metal.  I'll go through my so-called thought process, but if you just want the formulas, scroll to the bottom now.

I start with the top radius (Rtop), bottom radius (Rbot), and height (H) of the desired cone.

Start with these three dimensions

The first thing I do is figure out the angle of the cone, given all three measurements.  This was the key realization for me, because it's easy to make random cones that fit either the top OR the bottom, but the slope between the two sizes determines the entire layout.  Here's a cross section of half the cone.  We can draw a little right triangle with top dimension Rtop - Rbot, and height H.  The tangent of the angle (depicted by the theta symbol) is the opposite side length divided by adjacent side length (remember 10th grade?) so the angle can be determined by taking the inverse tangent of (Rtop-Rbot) / H.

Figure out the Cone's Angle
Now extend the bottom and center lines to where they meet below the cone.  This point is the center of the cutting pattern.  The theta angle is the same down here, and can be used to figure out the distances from the center to r1 and r2, the radii of the two arcs to be cut.  Again go back to basic trig and remember that the sine of an angle in a right triangle is the opposite side length over the hypotenuse length.  Sin theta = Rbot / r1, and the same for the larger triangle.  Rearrangement gives r1 = Rbot / sin theta and r2 = Rtop / sin theta.

Calculating the radii of the cuts
Now you have two radii and you can cut a couple of concentric circles.  The last part we need is the angle of the wedge we need to make that flat circle into a cone.  Start with the radius of the cone's top.  That gives the circumference = 2 x pi x Rtop.  So that's the length of the arc we need.  If we took our larger cutting radius (r2) and cut a full circle, it would be 2 x pi x r2 in length.  The ratio between those lengths is the amount of the full circle we need, but 2 x pi x Rtop / 2 x pi x r2 can be simplified to simply Rtop / r2, which will give the ratio we need.  You could also do this with the lower Rbot and r1 dimensions, which is a good check because it should be the same.  Multiplying that ratio by 360 degrees (in a full circle) gives the angle we need for our cutting pattern.

Determining the angle for the cut pattern

You will likely need to leave some material to overlap where the seam of the cone joins, for gluing / soldering / duct tape / whatever, and also some extra around the top and bottom to mate the cone to other things, so I recommend creating a scale mockup out of paper before you commit to anything big or expensive.

Here are the formulas for making a pattern for a funnel or cone out of flat stock.  Refer to the figures above for reference.

Rtop: radius of the top (wider) opening
Rbot: radius of the bottom opening
H: vertical height through centerline

first calculate theta: the cone's half-angle = tan-1((Rtop-Rbot) / H)

r1: radius of inner circle to cut = Rbot / sin(theta)
r2: radius of outer circle to cut = Rtop / sin(theta)
A: angle of arc = (Rtop / r2) x 360