Cylinder Area Paradox

Let be the surface of a cylinder of height and radius . ( does not include the flat circular ends of the cylinder.) This Demonstration constructs a set of triangles that tend uniformly to —yet their total area does not tend to the area of !
Divide into subcylinders (or bands) of height . Construct congruent isosceles triangles in each band with vertices at the vertices of a regular -gon inscribed in the circles at the top and bottom of each band, offset by .
For any point in (except the axis of the cylinder), let be the axial projection of onto . As , to say that the triangles approximate uniformly means that for any point on a triangle and any (independent of ), there is a such that for all , .
The sum of the areas of the triangles is
.
Depending on how the limit is taken, can differ. If first with held fixed and then , the limit is , the expected area of the cylinder. If first with held fixed and then , the limit is infinity. If and together so that is some positive constant , the limit can be chosen to be any number greater than .
Therefore does not have a limit.
The surface is known as Schwarz's lantern, Schwarz's polyhedron, or Schwarz's cylinder.

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For details, see Freida Zames' Surface Area and the Cylinder Area Paradox.
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