We define a polyhedral version of a stratification for projection maps that applies to any complex or real toric variety and show that it yields similarly desirable properties to the classical map stratification of a proper map. Our results are constructive and give rise to a method for associating the Whitney strata of the projection to the faces of the polytope of the corresponding toric variety. For all the examples we consider, our resulting algorithm outperforms known general purpose methods, e.g., Helmer and Nanda (FoCM, 2022), and Đinh and Jelonek (DCG, 2021), for computing map stratifications.