we present the tetrahedral compactness theorem which states that sequences of riemannian manifolds with a uniform upper bound on volume and diameter that satisfy a uniform tetrahedral property have a subsequence which converges in the gromov-hausdorff sense to a countably $\mathcal{h}^m$ rectifiable metric space of the same dimension. the tetrahedral property depends only on distances between points in spheres, yet we show it provides a lower bound on the volumes of balls. the proof is based upon intrinsic flat convergence and a new notion called the sliced filling volume of a ball.