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We investigate the relationship, in various contexts, between a closed geodesic with self‐intersection number k (for brevity, called a k‐geodesic) and its length. We show that for a fixed compact hyperbolic surface, the short k‐geodesics have length comparable with the square root of k. On the other hand, if the fixed hyperbolic surface has a cusp and is not the punctured disc, then the short k‐geodesics have length comparable with log k. The length of a k‐geodesic on any hyperbolic surface is known to be bounded from below by a constant that goes to infinity with k. In this paper, we show that the optimal constants {Mk} are comparable with log k leading to a generalization of the well‐known fact that length less than 4 log⁡(1+2) implies simple. Finally, we show that for each natural number k, there exists a hyperbolic surface where the constant Mk is realized as the length of a k‐geodesic. This was previously known for k = 1, where M1 is the length of the figure eight on the thrice punctured sphere.