There are two ways (among many others) to compute the volume of a (convex) polytope. One using a formula of Brion and another using an argument of P. McMullen and R. Schneider. The ensuing identity suggests a non-commutative generalization which we can currently prove for Coxeter zonotopes (e.g. a permutahedron). This algebraic equality plays a role in Arthur's trace formula. This has applications to spectral asymptotics of locally symmetric spaces. No prior knowledge of these subjects is assumed. Joint work with Tobias Finis and Werner Muller.