Systole length in hyperbolic n-Manifolds

We show that the length R of a systole of a closed hyperbolic n-manifold (n >= 3) admitting a triangulation by t n-simplices can be bounded below by a function of n and t, namely R >= 1/2((nt)O(n4t)). We do this by finding a relation between the number of n-simplices and the diameter of the manifold and by giving explicit bounds for a well known relation between the length of the core curve of a Margulis tube and its radius. We prove the same result for finite volume manifolds, with a similar but slightly more involved proof.