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|
|Calculating the radii of the cuts|
|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