Disregarding crest, spines, and rootball, if one were to model a cactus as non-convex prism (say a heptagrammic (7-2) prism for a seven ribbed cactus) how could one write an equation to predict changes in surface area as additional 'ribs', each of which is composed of two faces, are added to the prism? Here is the tricky part, in this cactus prism the core circumference where the ribs differentiate and the outer circumference where the ribs terminate must remain consistent regardless of rib count. I've provided an illustration that I hope clarifies what I am describing.
Or, for the sake of simplicity, a two dimensional model would be completely sufficient to describe a cross section of a columnar cactus. So how could one predict the changes in perimeter as additional points are added to a series of star polygons, each of which is inscribed upon a consistently sized annulus between two concentric circles? Any help writing such an equation would be greatly appreciated. Additionally, would one expect the increase in perimeter to be linear as additional points are added to the star polygon?
Thanks to Beavis for answering this question with the following equation, which has been adapted to describe the surface area of a non convex star prism sans polygon faces:
