The heights of two circular cylinders of equal volume are in the ratio 1 : 2.

(b) $\sqrt{2}: 1$

Let the radii of the two cylinders be r and R and their heights be h and 2h, respectively.
Since the volumes of the cylinders are equal, therefore:

$\pi \times r^{2} \times h=\pi \times R^{2} \times 2 h$

$\Rightarrow \frac{r^{2}}{R^{2}}=\frac{2}{1}$

$\Rightarrow\left(\frac{r}{R}\right)^{2}=\frac{2}{1}$

$\Rightarrow \frac{r}{R}=\frac{\sqrt{2}}{1}$

$\Rightarrow r: R=\sqrt{2}: 1$

Hence, the ratio of their radii is $\sqrt{2}: 1$.