  Given a polygon $P$, for two points $s$ and $t$ contained in the polygon,
their \emph{geodesic distance} is the length of the shortest $st$-path within
$P$. A \emph{geodesic disk} of radius $r$ centered at a point $v \in P$ is the
set of points in $P$ whose geodesic distance to $v$ is at most $r$. We present
a polynomial time $2$-approximation algorithm for finding a densest geodesic
unit disk packing in $P$. Allowing arbitrary radii but constraining the number
of disks to be $k$, we present a $4$-approximation algorithm for finding a
packing in $P$ with $k$ geodesic disks whose minimum radius is maximized. We
then turn our focus on \emph{coverings} of $P$ and present a $2$-approximation
algorithm for covering $P$ with $k$ geodesic disks whose maximal radius is
minimized. Furthermore, we show that all these problems are $\mathsf{NP}$-hard
in polygons with holes. Lastly, we present a polynomial time exact algorithm
which covers a polygon with two geodesic disks of minimum maximal radius.
