Say I start with the cone shown in the left in the diagram below. I can find the angle
a formed by the wall of the cone as
If I cut the cone open and flatten the wall into a 2D surface, it forms a segment of a circle having radius
S and an arc length equal to the circumference of the cone,
2πR. The segment angle
b can be found as the fraction of the circle comparing the segment arc to the circle circumference, which reduces to
If I want to calculate the actual angles, I need a trig table for the cone. If I'm only interested in the relationship between
b, I could calculate the angles and compare them. However, it intuitively seems like there might be some simple ratio or relationship between the two angles.
That's my question. Is there a simple ratio or relationship between the cone angle and the angle of its flattened surface?