I don't happen to own a planimeter, but there are simulated ones on the internet. These use Green's theorem to compute area of a traced curve - you trace out the curve and a gadget mechanically collects the dot products of a fixed vector field and your tangent vector. If you take your vector field to have constant curl, then the gadget has computed for you the area of a region. I have demonstrated this to calculus students after teaching them Green's theorem, and they find it impressive, generally.
Here is a link to one: http://www.hpmuseum.org/planim/planimtr.htm#the_applet . There is a link to a guide which describes how it works, but you may have to work it out yourself, since it's rather terse.
Another classic is to ask them to construct a Mobius strip out of duct tape, and to cut it down the middle, three levels deep (so 1 + 1 + 2 = 4 cuts in all, recording their observations. This serves as an enticement for vector calculus students to learn topology.