Can dis-similar polyhedral solids have the same surface area to volume ratio?

Question

I am trying to find an efficient way to determine if two polyhedral solids are similar or not. So I am wondering if dis-similar solids that have the same number of faces can have the same surface area to volume ratio? If that is impossible then I have my litmus test. :)

Answer 1

It won't work, sorry :-(

Take for example a big cube with a smaller cube glued onto one of the faces. You can slide the small cube around on the face to get dissimlar polyhedra without changing surface area or volume. This idea can also be done with many other shapes.