The radius of gyration $K$ of a body of mass $M$ and moment of inertia $I$ is given by $K = \sqrt{\frac{I}{M}}$.

For a circular disc of mass $M$ and radius $R$ about its respective axis (passing through the centre and perpendicular to its plane), the moment of inertia is $I_{disc} = \frac{1}{2}MR^2$. Its radius of gyration is $K_{disc} = \sqrt{\frac{\frac{1}{2}MR^2}{M}} = \frac{R}{\sqrt{2}}$.

For a circular ring of mass $M$ and radius $R$ about its respective axis, the moment of inertia is $I_{ring} = MR^2$. Its radius of gyration is $K_{ring} = \sqrt{\frac{MR^2}{M}} = R$.

The ratio of the radii of gyration is: $\frac{K_{disc}}{K_{ring}} = \frac{\frac{R}{\sqrt{2}}}{R} = \frac{1}{\sqrt{2}} = 1:\sqrt{2}$.