- We design a method of decomposing convex polytopes into simpler polytopes. This decomposition yields a way of calculating exactly the volume of the polytope, or, more generally, multiple integrals over the polytope, which is equivalent to the way suggested in , based on Fourier-Motzkin elimination ([10, pp. 155-157]). Our method is applicable for finding uniform decompositions of certain natural families of polytopes. Moreover, this allows us to find algorithmically an analytic expression for the distribution function of a random variable of the form ∑d i=1 ciXi, where (X1, X2, ..., Xd) is a random vector, uniformly distributed in a polytope.