  Given a connected graph $G$ on $n$ vertices, with minimum degree $\delta\geq
2$ and girth at least $g \geq 4$, what is the maximum radius $r$ this graph can
have? Erd\H{o}s, Pach, Pollack and Tuza established in the triangle-free case
($g=4$) that $r \leq \frac{n-2}{\delta}+12$, and noted that up to the value of
the additive constant, this is tight. We determine the exact value for the
triangle-free case. For higher $g$ little is known. We settle the order of $r$
for $g=6,8,12$ and prove an upper bound to the order for general even $g$.
Finally, we show that proving the corresponding lower bound for general even
$g$ is equivalent to the Erd\H{o}s girth conjecture.
