The polygonal distributions are a class of distributions that can be defined via the mixture of triangular distributions over the unit interval. The class includes the uniform and trapezoidal distributions, and is an alternative to the beta distribution. We demonstrate that the polygonal densities are dense in the class of continuous and concave densities with bounded second derivatives. Pointwise consistency and Hellinger consistency results for the maximum likelihood (ML) estimator are obtained. A useful model selection theorem is stated as well as results for a related distribution that is obtained via the pointwise square of polygonal density functions.