The ["napkin-ring problem"](http://en.wikipedia.org/wiki/Napkin_ring_problem) sometimes shows up in 2nd-year calculus courses, but it can fit quite neatly into a high-school geometry course via [Cavalieri's principle](http://en.wikipedia.org/wiki/Cavalieri%27s_principle).

However, the conclusion remains astonishing.  Is there some advanced viewpoint from which it becomes obvious from some sort of symmetry that's not visible in the naive formulation?