The ratio of radii of gyration of a circular ring and a circular disc, of the same mass and radius, about an axis passing through their centres and perpendicular to their planes are

Let M and R be mass and radius of the ring and the disc respectively. Then, 

Moment of inertia of ring about an axis passing through its centre and perpendicular to its plane is

$I_{\text{ring }}=M{R}^2$
Moment of inertia of disc about the same axis is

$I_{\mathrm{disc}}=\frac{M{R}^2}{2}$

$\text{ As }I=M{k}^2\text{ where }k\text{ is the radius of gyration }$

$$\begin{gathered} \therefore I_{\text{ring }}=M{k}_{\text{ring }}^2=M{R}^2 \\ \text{ or } k_{\text{ring }}=R \\ \text{ and } I_{\text{disc }}=M{k}_{\text{disc }}^2=\frac{M{R}^2}{2} \\ \text{ or } k_{\text{disc }}=\frac{R}{\sqrt{2}} \\ \therefore \frac{k_{\text{ring }}}{k_{\text{disc }}}=\frac{R}{R/\sqrt{2}}=\frac{\sqrt{2}}{1} \\ {k}_{\text{ring }}:k_{\text{disc }}=\sqrt{2}:1 \end{gathered}$$