  We apply a method developed by one of the authors, see \cite{Arl1}, to
localize the numerical range of \textit{quasi-sectorial} contractions
semigroups. Our main theorem establishes a relation between the numerical range
of quasi-sectorial contraction semigroups $\{\exp(- t S)\}_{t\ge 0}$, and the
maximal {sectorial} generators $S$. We also give a new prove of the rate
$O(1/n)$ for the operator-norm Euler formula approximation: $\exp(- t
S)=\lim\limits_{n\to \infty}(I+tS/n)^{-n}$, $t\ge 0$, for this class of
semigroups.
