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 $\mathfrak{m}^kH_{\mathfrak{m}}^1(R/I) = 0$. From that one can describe all arithmetically CohenMacaulay or Buchsbaum tetrahedral curves.