We propose a non-linear relationship between two of the most important measures of centrality in a network: degree and closeness. Based on a shortest-path tree approximation, we give an analytic derivation that shows the inverse of closeness is linearly dependent on the logarithm of degree. We show that our hypothesis works well for a range of networks produced from stochastic network models including the Erdos-Reyni and Barabasi-Albert models. We then test our relation on networks derived from a wide range of real-world data including social networks, communication networks, citation networks, co-author networks, and hyperlink networks. We find our relationship holds true within a few percent in most, but not all, cases. We suggest some ways that this relationship can be used to enhance network analysis.