Let \(n_ 1,\dots,n_ i\) be a sequence of positive integers. Fixing \(k\) and \(i\), one can consider all group presentations \(G = \langle a_ 1,\dots,a_ k;r_ 1,\dots,r_ i\rangle\) where \(r_ 1,\dots,r_ i\) are reduced words such that the length of \(r_ j\) is \(n_ j\) for \(j=1,\dots,i\). Let \(N = N(k,i,n_ 1,\dots,n_ i)\) be the number of all such presentations, \(N_ h\) be the number of hyperbolic presentations among these, and \(n = \min(n_ 1,\dots,n_ i)\). The author uses van Kampen diagrams to prove that \(\lim_{n\to \infty}N_ h/N = 1\), a claim of M. Gromov [Essays in group theory, Publ., Math. Sci. Res. Inst. 8, 75-263 (1987; Zbl 0634.20015)]. Furthermore, it is shown that the limit tends to 1 faster than some function \(1-\exp(cn)\), where \(c = c(i) < 0\).