  A widely used approximation of the Gaussian curvature on a triangulated
surface is the angle defect, which measures the deviation between $2\pi$ 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
$H^k$-Sobolev norms of integer order $k \ge -1$.
