On cycles in graphs with specified radius and diameter

Let $G$ be a graph of radius $r$ and diameter $d$ with $d\leq 2r-2$. We show that $G$ contains a cycle of length at least $4r-2d$, i.e. for its circumference it holds $c(G)\geq 4r-2d$. Moreover, for all positive integers $r$ and $d$ with $r\leq d\leq 2r-2$ there exists a graph of radius $r$ and diameter $d$ with circumference $4r-2d$.