Another way to think of the "spread" of an angle is the area of a unit rhombus with that angle.
Reminds me of David Hestenes and Geometric Algebra for Physics...
Nice geometrical picture, but why wouldn't "sine" be more appropriate for that? The area of a rhombus is equal to the length of a side times the perpendicular distance to the other side. If the rhombus is sitting with the 'angle' in question at the origin and one side along the x axis, then the y coordinate of the other side will be the perpendicular distance in question. And what's the y coordinate? Seems like it should be the sine of the angle.