Using geometric probability, we apply the formal definitions of Shannon entropy and Rényi's generalization to study the complexity of planar curves of finite length within a convex set. The bounds for the Shannon and Rényi entropies depend on the arc length of the curve and on that of the boundary of the convex set; they involve a Gibbs distribution and a power law distribution, respectively. We also obtain explicit formulae for the two entropies and determine convex sets that maximize the entropy of curves.