Here's a proof of the area of a circle (or sector) which is different from the one posted previously.
EDIT: I was unable to embed the file, which is in pdf form. Here is a link:
http://wildpositron.files.wordpress.com/2011/04/sectorarea2.pdf
I discussed what goes into making the proof complete to show that the map preserves area on my blog here (it requires just another picture or two, but it's essentially still only a geometric argument):
http://wildpositron.wordpress.com/2011/04/05/calculating-the-area-of-a-sector/
