(Very elementary stuff.) When n>2d, (d,n)=1, connecting every dth vertex of a regular n-gon makes an n-pointed star {n/d} consisting of d-2 concentric rings of kites surrounding a ring of isosceles triangles surrounding a regular n-gon. Working inward, successive rings of kites have winding numbers W=1,2,3,...; the triangles have W=d-1, and for the polygon, W=d. For unit edges, the area is %pi d cot(-----) n n n A(-) = ------------, d 4 where the component polygons are weighted by their winding numbers, i.e, W dx dy integrated over the whole plane. This correctly gives n n A(-----) = - A(-), n - d d i.e., mirror imaging negates the area by winding clockwise. The %pi d csc(-----) n circumradius = ----------, and the 2 %pi d cot(-----) n inradius = ----------. 2 For unit circumradius instead of unit sides, the area is %pi d sin(-----) n n n a(-) = ------------, d 2 which, for fixed n is clearly maximal for d=n/4. I.e., the star with greatest weighted area is the one with most nearly right-angled vertices, since the vertex angles are pi - 2 pi d/n. E.g., the star of Lakshmi (two squares) is maximal among the octagrams. Removing just the outermost edges that form the vertex angles leaves an {n/(d-1)} star polygon, if when (n,d-1)=g we interpret this as g {n/g,(d-1)/g} stars superposed. Interestingly, for such improper stars, continued edge deletion recovers the proper ones with lesser d. E.g., {9/4} -> {9/3} (star of Goliath = three triangles) -> {9/2}. This tells us all the interior angles. The circumradius formula for consecutive d provides the proportions of the inner stars, and hence the sizes of all the kites, etc. Note also that the area formulas work for improper stars: A((gn)/(gd)) = g A(n/d). --rwg I keep warm peanut butter in my pockets AND I VOTE