Intersection bodies are classical objects from convex geometry, that are constructed from given convex body. In the past, they have mainly been studied from the point of view of convex analysis. In this talk we investigate combinatorial and algebraic structures of intersection bodies of polytopes. We consider an algorithm to compute both the radial function and the algebraic boundary of these intersection bodies, and provide an upper bound for their degree. This is joint work with Katalin Berlow, Chiara Meroni and Isabelle Shankar.