The goal of this talk is to show how tools from topology can bound or compute quantities arising in metric geometry. I’ll begin by introducing the Hausdorff and Gromov-Hausdorff distances, which are ways to measure the “distance” between two metric spaces. Though Hausdorff distances are easy to compute, Gromov-Hausdorff distances are not. Next I will explain the nerve lemma, which says when a cover of a space faithfully encodes the shape of that space. As the main result, I’ll show how when X is a sufficiently dense subset of a closed Riemannian manifold M, we can use the nerve lemma to lower bound the Gromov-Hausdorff distance between X and M by 1/2 the Hausdorff distance between them. The constant 1/2 can be improved, and even obtains the optimal value 1 (meaning the Hausdorff and Gromov-Hausdorff distances coincide) when M is the circle.