What has an elbow, a wheel, and a pivot, and can deduce area from arclength? Answer: an elegantly simple device called a planimeter. How the planimeter works can be explained using basic geometry, or, for the adventurous reader whose mind is ringing with "Green's Theorem", using calculus (see my article in the College Mathematics Journal, January 2007). I greatly recommend some hands-on playing with these wonderful devices, if you can find one. The pictures below are two of my planimeters (I currently have 6 in my collection).
Another website with a nice mathematical explanation.
Professor Robert L. Foote of Wabash College has an extensive webpage on planimeters.
Bruce Atwood has a planimeter site with a proof using the trapeziodal rule.
You can buy your very own planimeter from Vintage Instruments or from private sellers via eBay.