Given a pure $d$-dimensional simplicial complex embedded in $\mathbb{R}^d$, we may measure the volumes of its maximal simplices. We say such an embedding is $d$-volume rigid if the only continuous deformations of its vertices are images of the whole complex under a $d$-volume preserving transformation, and that the simplicial complex is $d$-volume rigid if a typical embedding of it is $d$-volume rigid. In Euclidean bar-joint rigidity, Maxwell counts provide the least number of edges required for rigidity in $\mathbb{R}^d$ in terms of $d$ and the number of vertices. In a similar spirit, we provide the least number of $k$-simplices, for all $0\leq k\leq d$, required for a complex to be $d$-volume rigid in terms of $d$ and the number of vertices. We use recent applications of algebraic combinatorics to the volume rigidity problem, as well as what is known about the volume rigidity matroid.