  Let $k,d $ be positive integers. We determine a sequence of constants that
are asymptotic to the probability that the cluster at the origin in a
$d$-dimensional Poisson Boolean model with balls of fixed radius is of order
$k$, as the intensity becomes large. Using this, we determine the asymptotics
of the mean of the number of components of order $k$, denoted $S_{n,k}$ in a
random geometric graph on $n$ uniformly distributed vertices in a smoothly
bounded compact region of $R^d$, with distance parameter $r(n)$ chosen so that
the expected degree grows slowly as $n$ becomes large (the so-called mildly
dense limiting regime). We also show that the variance of $S_{n,k}$ is
asymptotic to its mean, and prove Poisson and normal approximation results for
$S_{n,k}$ in this limiting regime. We provide analogous results for the
corresponding Poisson process (i.e. with a Poisson number of points).
  We also give similar results in the so-called mildly sparse limiting regime
where $r(n)$ is chosen so the expected degree decays slowly to zero as $n $
becomes large.
