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Let $G$ be a graph of radius $r$ and diameter $d$ with $d\leq 2r-2$. We showthat $G$ contains a cycle of length at least $4r-2d$, i.e. for itscircumference 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 $r$ anddiameter $d$ with circumference $4r-2d$.

On cycles in graphs with specified radius and diameter