A widely used approximation of the Gaussian curvature on a triangulated surface is the angle defect, which measures the deviation between and the sum of the angles between neighboring edges emanating from a common vertex. We show that the linearization of the angle defect about an arbitrary piecewise constant Regge metric is related to the classical Hellan--Herrmann--Johnson finite element discretization of the div div operator. Integrating this relation leads to an integral formula for the angle defect which is well-suited for analysis and generalizes naturally to higher order. We prove error estimates for these high-order approximations of the Gaussian curvature in -Sobolev norms of integer order .