It is shown that the diameter $\mathrm{diam}(H_{\mathfrak{m}}^1(R/I))$ of the first local cohomology module of a tetrahedral curve $C=C(a_1,\dots ,a_6)$ can be explicitly expressed in terms of the ai and is the smallest non-negative integer k such that ${\mathfrak{m}}^k{H}_{\mathfrak{m}}^1(R/I)=0$. From that one can describe all arithmetically CohenMacaulay or Buchsbaum tetrahedral curves.