jmc

Since $\angle RIP$ is inscribed in arc $RP$, the measure of arc $RP$ is $2\angle RIP=72^{\circ }$. Therefore, arc $RP$ is $\frac{72}{360}=\frac{1}{5}$ of the circumference of the whole circle. The circle's circumference is $2OR\pi =20\pi$ cm, so the length of arc $RP$ is $\frac{1}{5}\cdot 20\pi =\boxed{4\pi }$ cm.