Circular data clustering has recently been solved exactly in sub-quadratic time. However, the solution requires a given number of clusters; methods for choosing this number on linear data are inapplicable to circular data. To fill this gap, we introduce the circular silhouette to measure cluster quality and a fast algorithm to calculate the average silhouette width. The algorithm runs in linear time to the number of points on sorted data, instead of quadratic time by the silhouette definition. Empirically, it is over 3000 times faster than by silhouette definition on 1,000,000 circular data points in five clusters. On simulated datasets, the algorithm returned correct numbers of clusters. We identified clusters on round genomes of human mitochondria and bacteria. On sunspot activity data, we found changed solar-cycle patterns over the past two centuries. Using the circular silhouette not only eliminates the subjective selection of number of clusters, but is also scalable to big circular and periodic data abundant in science, engineering, and medicine.