Can we approximate (arbitrarily closely) a compact $C^1$ hypersurface in Euclidean space without boundary with a polygonal hypersurface, such as a simplicial complex? To clarify, I want to have the $\mathcal{H}^{n-1}$ area of the approximation converge to that of the hypersurface as the diameter of the polygons goes to 0.

If so, can you please provide a reference for the theorem. Thank you kindly.

Note, I know this to be true for $C^2$ since we have positive reach and can create a tubular neighborhood with the normals.

Edited: I think I made a mistake to say "same size" above (although that would be ideal for my situation) so I removed it. So in the case of allowing different sizes, can it be done? In fact, I've looked more carefully and they wouldn't have to be the same shape either. The critical piece is to have a piecewise linear surface composed of $(n−1)$-dimensional polyhedra, meaning the surface is continuous with no "breaks" or "jumps" at the edges, since I will integrate over each piece. Also, there should be a finite number of pieces since the surface is compact, correct? Thanks!