Surface Area of a Hypercube I am interested in computing the surface area of an $n$-dimensional hypercube and am interested in a reference or an answer which defines the notion of surface area for higher dimensional polytopes as I am trying to compute the surface area of an infinite family of duoprisms and knowing the surface area of an $n$-dimensional hypercube would be very useful to my understanding.
That is, what is the surface area of an $n$-dimensional hypercube with side length $s$, and how can you think about surface area of higher dimensional polytopes in general?
EDIT: In regarding as to whether I am referring to "surface area" or "surface volume", I am interested in understanding any $k$-dimensional version of surface hyper(area/volume) for an $n$-dimensional polytope. I think it makes sense that the $n$-dimensional version of this quantity is the volume, the $(n-1)$-dimensional version would be surface "area", and there are $(n-2),...,1$-dimensional versions of this idea. Is there any way to understand this with differential forms perhaps?

Expand the polynomial $(x+2)^n$.  For example, $(x+2)^3=1x^3+6x^2+12x+8$.  Read off the coefficients: A cube has 1 cube, 6 faces, 12 edges and 8 corners.  
