What is the term for the number of sides of a polygon? Or more generally the number of (n-1)-polytopes forming a n-polytope?
We have 'cardinality' for the number of elements in a set. In the domain of topology, 'valence' is the number of edges connecting to a vertex. Etc.
For polygons, it seems to always be 'the number of sides'. Isn't there a word for it?
 A: An $n$-gon is a polygon with $n$ sides; for example, a triangle is a 3-gon. A polygon is a 2-dimensional example of the more general polytope in any number of dimensions.

Also see here. Hope it helps. 
A: In the theory of polytopes, faces, edges and vertices are generalised by the terms $elements$. So in a $3$-polytope (polyhedron), the dimension $2\ (=3-1)$ elements are called faces, while the dimension $1\ (=3-2)$ elements are called edges and dimension $0\ (=3-3)$ elements are called vertices. There will be elements of other dimensions for hyperdimensional polytopes.
To the best of my knowledge, there is no other generally accepted term for what you are looking for. I have seen the word "arity" used occasionally to refer to the number of sides of a polygon ($2$-polytope), but I think this is an uncommon usage and would need to be defined prior to being introduced in a text or the meaning should be crystal clear in the context.
A: I'd say "number of faces". For polygons with $n$ edges and $n$ vertices, the number $n$ plays a unique role and one might call $n$ something like the "order" or "degree" of that polygon (or "arity", see Deepak's answer). But for polyhedra and higher polytopes? There we have different counts of vertices, edges, faces, etc., so that the mere number of maximal-dimensional elements seems to be less classifying in nature. In fact, we tend to introduce and prefer classifying numbers that are not immediately among the element counts, such as genus. So to repeat, "number of faces" (or "number of $(n-1)$-faces") seems to be most adequate.
