In this paper, we present new analytical and numerical modal analyses of bent planar waveguides allowing precise computation of the bending loss. The analytical method is based on an accurate algorithm for evaluating Bessel and Hankel functions. The numerical one relies on a finite-element model derived from the cylindrical coordinates version of the Maxwell system. To eliminate spurious reflections from the artificial boundary of the computational domain, an efficient perfectly matched layer technique is adapted to the formulation. A thorough comparison shows that our analytical approach is both simple and highly accurate whereas the numerical method is cheap and easily extendible to higher dimensions and other geometries.