The areas of a sequence of polyhedra approaching a surface need not approach the area of the surface, but there are theorems guaranteeing that this be so. (T. Rado, On the Problem of Plateau, Chapter 1.)
What is known about analogous situations for higher dimensional manifolds in $R^n$? I'd like an answer in terms that an advanced undergraduate can understand.
Check out my old paper with the late, lamented Fred Almgren, and especially the references there in (e.g. to the paper of Allard). The magic word is "varifold". On second thought, if you look at Frank Morgan's Geometric Measure Theory book, there is something called the Approximation Theorem, which might be just what you need. (page 77 is the beginning of the chapter on the subject, in the fourth edition of Morgan's book).
