Define dimensions

For a sphere of radius $r$ inscribed in a cylinder, the radius of the cylinder is also $r$, and the height of the cylinder is equal to the diameter of the sphere, so $h = 2r$.

$h = 2r$

Calculate volumes

The volume of the sphere is $V_s = \frac{4}{3}\pi r^3$ and the volume of the cylinder is $V_c = \pi r^2 (2r) = 2\pi r^3$.

$V_s = \frac{4}{3}\pi r^3, V_c = 2\pi r^3$

Find the ratio

The ratio of the volume of the sphere to the volume of the cylinder is $\frac{\frac{4}{3}\pi r^3}{2\pi r^3} = \frac{4/3}{2} = \frac{2}{3}$.

$\text{Ratio} = 2:3$