  We develop two simple and efficient approximation algorithms for the
continuous $k$-medians problems, where we seek to find the optimal location of
$k$ facilities among a continuum of client points in a convex polygon $C$ with
$n$ vertices in a way that the total (average) Euclidean distance between
clients and their nearest facility is minimized. Both algorithms run in
$\mathcal{O}(n + k + k \log n)$ time. Our algorithms produce solutions within a
factor of 2.002 of optimality. In addition, our simulation results applied to
the convex hulls of the State of Massachusetts and the Town of Brookline, MA
show that our algorithms generally perform within a range of 5\% to 22\% of
optimality in practice.
