This paper deals with the problem of estimating 2D shape complexity. This has important applications in computer vision as well as in developing efficient shape classification algorithms. We define shape complexity using correlates of Kolmogorov complexity - entropy measures of global distance and local angle, and a measure of shape randomness. We tested our algorithm on synthetic and real world datasets with excellent results. We also conducted user studies that indicate that our measure is highly correlated with human perception. They also reveal an intuitive shape sensitivity curve - simple shapes are easily distinguished by small complexity variations, while complex shapes require significant complexity differences to be differentiated.