Products of Diagonal Lengths in the Regular Polygon

A regular polygon is scaled up in size so that its side length equals the length of some diagonal from the smaller polygon (the magnification diagonal). The diagonal lengths in the large polygon are seen to be sums of those in the smaller. The illustration is for an 11-gon.

The diagonals are numbered in order of increasing size. Choose a magnification diagonal and a diagonal from the larger polygon. Then move the slider to see the small diagonals involved in the sum.

If diagonal is scaled up using , then . The proof of this formula depends upon the fact that if , then the diagonal in a polygon of unit radius has length .