  We describe the configuration space $\mathbf{S}$ of polygons with prescribed
edge slopes, and study the perimeter $\mathcal{P}$ as a Morse function on
$\mathbf{S}$. We characterize critical points of $\mathcal{P}$ (these are
\textit{tangential} polygons) and compute their Morse indices. This setup is
motivated by a number of results about critical points and Morse indices of the
oriented area function defined on the configuration space of polygons with
prescribed edge lengths (flexible polygons). As a by-product, we present an
independent computation of the Morse index of the area function (obtained
earlier by G. Panina and A. Zhukova).
