A vector partition function is the number of ways to write a vector as a non-negative integer-coefficient sum of the elements of a finite set of vectors $Δ$. We present a new algorithm for computing closed-form formulas for vector partition functions as quasi-polynomials over a finite set of pointed polyhedral cones, implemented in the ``calculator'' computer algebra system. We include an exposition of previously known theory of vector partition functions. While our results are not new, our exposition is elementary and self-contained.