The ratio between the curved surface area and the total surface area of a right circular cylinder is 1 : 2.

Suppose that the curved surface area and the total surface area of the right circular cylinder are x cm2 and 2x cm2.
Then we have:
2x = 616
x = 308 sq cm
Hence, the curved surface area of the cylinder is 308 sq cm.
Let  r cm and h cm be the radius and height of the cylinder, respectively.

Then $2 \pi r h+2 \pi r^{2}=616 \mathrm{~cm}^{2}$ and $2 \pi r h=308 \mathrm{~cm}^{2}$

$\therefore 2 \pi r h+2 \pi r^{2}-2 \pi r h=616-308$

$\Rightarrow 2 \pi r^{2}=308$

$\Rightarrow 2 \times \frac{22}{7} \times r^{2}=308$

$\Rightarrow r^{2}=\frac{308 \times 7}{44}=49$

$\Rightarrow r=7 \mathrm{~cm}$

Now, $2 \pi r h=308 \mathrm{~cm}^{2}$

$\Rightarrow 2 \times \frac{22}{7} \times 7 \times h=308$

$\Rightarrow h=\frac{308}{44}=7 \mathrm{~cm}$

$\therefore$ Volume of the cylinder $=\pi r^{2} h$ cubic $\mathrm{cm}$

$=\frac{22}{7} \times 7^{2} \times 7$

$=1078 \mathrm{~cm}^{3}$