In this brief video, Minute Physics shows a visual proof demonstrating why ?R² gets you the area of a circle. By using various lengths of chain laid into a circular shape, you can make a triangle from the circle -- and from there it's a hop, skip, and a trigonometry jump to proof of your circle's area. Check it:

As Minute Physics explains in the YouTube video description, this video is based on a paper by Russell Jay Hendel of Dowling College: Proof Without Words: Area of a Disk is ?R² (PDF link).

See also: Explaining E=mc² Using a Radioactive Space Cat.