Why cone development matters
A cone or frustum, like a hopper, funnel, or transition duct, has to start life as a flat piece of sheet metal before it gets rolled and welded into shape. Getting that flat pattern wrong means a seam that does not close, a part that is too short or too tall once rolled, or wasted material on an expensive sheet.
How the numbers are calculated
Slant height is the straight-line distance along the cone surface from base to top, found using the Pythagorean theorem from the height and the difference between base and top radius. The development sector angle is the arc angle of the flat pattern once unrolled, scaled from a full 360 degree circle down to match the base circumference. Setting top radius to zero develops a full cone rather than a frustum with a flat top.
Practical tips
Always add a seam allowance to the flat pattern width before cutting, since this calculation gives the theoretical development with no overlap for welding or fastening. Mark the outer radius arc first, then the sector angle from a fixed center point, and check the finished layout against a physical mockup on cardboard before committing to metal on anything unusual or expensive.