  We analyze finite element discretizations of scalar curvature in dimension $N
\ge 2$. Our analysis focuses on piecewise polynomial interpolants of a smooth
Riemannian metric $g$ on a simplicial triangulation of a polyhedral domain
$\Omega \subset \mathbb{R}^N$ having maximum element diameter $h$. We show that
if such an interpolant $g_h$ has polynomial degree $r \ge 0$ and possesses
single-valued tangential-tangential components on codimension-1 simplices, then
it admits a natural notion of (densitized) scalar curvature that converges in
the $H^{-2}(\Omega)$-norm to the (densitized) scalar curvature of $g$ at a rate
of $O(h^{r+1})$ as $h \to 0$, provided that either $N = 2$ or $r \ge 1$. As a
special case, our result implies the convergence in $H^{-2}(\Omega)$ of the
widely used "angle defect" approximation of Gaussian curvature on
two-dimensional triangulations, without stringent assumptions on the
interpolated metric $g_h$. We present numerical experiments that indicate that
our analytical estimates are sharp.
