I think I mentioned at some point that it was easy to make an elegant pentagon from a circle of paper. It is, but still, it’s not intuitive. Making polygons from circles has a lot in common with compass-and-straightedge work. Except paper is easier to work with and the straightedge can be marked up like crazy.
Based on a method described by H.W. Richards in 1893 – not that I’ve seen this. I read about it at MathWorld.