I will discuss tropical methods for computing geometric invariants of semialgebraic sets, defined by polynomial equations and inequalities on valuations over an algebraically closed valued field, through relations to motivic integration in the sense of Hrushovski-Kazhdan, in which the universal invariant is expressed in terms of a tensor product of a scissors ring of polyhedra with a Grothendieck ring of varieties over the residue field. Based on joint work with Nicaise and Schroeter.