To compute the hyperbolicity constant is an almost intractable problem, thus it is natural to try to bound it in terms of some parameters of the graph. Let $\mathcal{G}(g,c,n)$ be the set of graphs $G$ with girth $g(G)=g$, circumference $c(G)=c$, and $n$ vertices; and let $\mathcal{H}(g,c,m)$ be the set of graphs with girth $g$, circumference $c$, and $m$ edges. In this work, we study the four following extremal problems on graphs: $A(g,c,n)=\min\{δ(G)\,|\; G \in \mathcal{G}(g,c,n) \}$, $B(g,c,n)=\max\{δ(G)\,|\; G \in \mathcal{G}(g,c,n) \}$, $α(g,c,m)=\min\{δ(G)\,|\; \in \mathcal{H}(g,c,m) \}$ and $β(g,c,m)=\max\{δ(G)\,|\; G \in \mathcal{H}(g,c,m) \}$. In particular, we obtain bounds for $A(g,c,n)$ and $α(g,c,m)$, and we compute the precise value of $B(g,c,n)$ and $β(g,c,m)$ for all values of $g$, $c$, $n$ and $m$.