I like your method of calculating facet area and dividing the total area by the facet area. It's simple and you don't have to count too much. The question is, how accurate do you want this measurement of the number of facets to be?
Maybe you can measure the area of several different facets, take the average, and divide the sphere area by this area. You should be careful to include half the width of the facet wall. Say if the thickness of a wall is and the distance between two walls is , the area of the facet is . The correction is , and this is the effective area of the facet in your measurement.
This is all I can say on this for now.