Mamikon Mnatsakanian’s Idea of Sectors on Polygon

I came across this on Starbird’s Lecture on Visual Geometry. What Mnatsakanian idea does is to draw sectors of same radius on the extended side of the polygon (It need not be regular). The benefit of this is we get a full circle when we re-arrange these sectors as evident from the figure below. However the amazing thing happens is when the number of sides of the polygon increased to infinity and in that case this circle is annulus