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It turns out that it is often easier to solve cylindrical surface area problems in spherical coordinates. I don't know why. A really fascinating calculation is to find the surface area of a slice of a sphere, say between the 45th and 50th parallels and to find the surface area of a slice (at the same height) of the circumscribed cylinder of the same radius. It turns out that these two areas are always the same–a fact surely known to cartographers.