For a positive integer $s$ and for $0\le a< b,$ let $$K=K^s_{a,b}=\bigcup_{k=0}^{s-1}e^{2\pi i\frac{k}{s}}[a,b].$$
We find that the {\it capacity} $\mathrm{Cap} (K)$ of $K$ is
$$\mathrm{Cap} (K) = \sqrt [s]{\frac{b^s-a^s}4}\cdot\eqno (1)$$
\par From this relation we derive several classical results, due to Akhiezer, Henrici, and Bartolomeo and He, on capacities of some sets
in the complex plane.
\par An extension relation (1) to more general sets in the complex plane,
together with potential theoretic techniques, is then used to
obtain {\it saturation} theorems pertaining to approximation
by polynomials with integer coefficients.