One of the goals of modern algebraic geometry is to gain a better understanding of the intersection theory of the moduli space of stable pointed curves. In this talk, I will discuss a new perspective concerning a certain class of intersection numbers on these moduli spaces. It turns out that one can encapsulate all of these intersection numbers as evaluations of Ehrhart polynomials, which are counting functions for the number of lattice points inside dilates of polyhedral objects. This allows one to state a simple enumerative interpretation of these intersection numbers, which was not known before. The talk will be tailored to a general algebro-geometric audience, so no background in moduli of curves or Ehrhart theory will be assumed.