TITLE: Naive definition of surface area doesn't work?
QUESTION [12 upvotes]: A first stab at a definition of surface area might go like this:
Let S be a surface.  Select finitely many points from S and make a bunch of triangles having these points as vertexes. Add up the areas of all of the triangles. This is a finite approximation to the surface area.  Now to get the actual surface area, we increase the number of points so that they "densely cover" the surface (Maybe a good working definition would be that any open set of S should eventually contain some vertexes of triangles).
I seem to remember reading about a counterexample to this naive definition, but I can't find a reference.  I believe there is even a natural looking polygonal approximation to the cylinder whose surface area diverges to infinity.  Can anyone help me out?

REPLY [25 votes]: Perhaps you are thinking of the Schwartz Lantern? 
It converges to the cylinder in the Hausdorff metric but its area can be arranged
to head toward $\infty$.
It was mentioned in the earlier MO question, "Convergence of finite element method: counterexamples."
There is nice applet here showing the lantern rotating.
Here is an image from Conan Wu's blog: