# What is the largest possible value of the smallest angle between the diagonals of an $n$-gon for even $n$?

In this answer, I showed that the maximum value of the smallest angle between two diagonals of a $$21-$$gon is $$8\frac{4}{7}$$ degrees. The method given generalizes to all polygons with an odd number of sides, other than $$3,$$ giving an angle of $$180/n$$ degrees. I also think that $$180/n$$ is optimal for even numbers other than $$4,$$ since a regular polygon achieves this angle. For $$4,$$ the optimal angle is $$90$$ degrees, which is achieved by a square. What is the largest possible value of the smallest angle between the diagonals of an $$n$$-gon for even $$n$$?

We can look at the diagonals forming one or two stars which connect two vertices $$n-1$$ edges away if the number of vertices of the polygon is $$2n$$ with $$n > 2.$$ Apart from pairs of such diagonals, every diagonal intersects the other diagonals either at a vertex or inside the polygon. Therefore, these account for at least $$n$$ directions of diagonals. To find $$n$$ more, we can look at the diagonals connecting two vertices $$n$$ edges away. Since all these diagonals intersect each other inside the polygon and the previous diagonals either at a vertex or inside the polygon, there are $$n$$ new directions formed. Therefore, there are at least $$2n$$ directions of diagonals, so the minimum angle between diagonals is at most $$180/2n.$$