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. The length of a $k$-geodesic on a 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 $M_k$ are comparable to $ \log k$. A consequence is a generalization of a well-known fact that a closed geodesic of length less than $4 \log (1+\sqrt 2)$ is simple. Namely, a closed geodesic of length less than $1/4 \log (2 k)$ ($k$ a natural number) has at most $k$ self-intersections.
Finally, we show that for each natural number $k$, there exists a hyperbolic surface where the constant $M_k$ is realized as the length of a $k$-geodesic. This was previously known for $k = 1$, where the figure 8 on the thrice punctured sphere is the shortest non-simple closed geodesic.