4. (10 points) There is a square timber and a cylindrical timber of the same height, with a volume ratio of 1:1. If the square timber is processed into the largest possible cylinder, and the cylindrical timber is processed into the largest possible rectangular prism, then the ratio of the volume of the cylinder to the volume of the rectangular prism is $\qquad$ .
【Analysis】The volume and height of the square timber and the round timber are equal, which means the base area is also equal. To find the ratio of the volume of the cylinder processed from the round timber to the volume of the cuboid processed from the square timber, it is sufficient to compare the base areas. Therefore, we only need to find the base areas, and then consider the inscribed circle and circumscribed circle of a square.

【Solution】Solution: (1)

Let the radius of the circle be $r$, the ratio of the area of the circle to the area of the square is:
$\left(\pi r^{2}\right):(2 r \times 2 r)=\frac{\pi}{4}$,
(2)

Let the radius of the circle be $r$, the ratio of the area of the square to the area of the circle is:
$$
(2 r \times r):\left(\pi \times r^{2}\right)=\frac{2}{\pi},
$$

Since the volume and height of the square timber and the round timber are equal, it means the base area is also equal, i.e., the area of the large square in figure (1) is equal to the area of the large circle in figure (2),

Therefore, the ratio of the volume of the cylinder to the volume of the cuboid is:
$$
\frac{\pi}{4}: \frac{2}{\pi}=\frac{\pi^{2}}{8} ;
$$

Answer: The ratio of the volume of the cylinder to the volume of the cuboid is $\frac{\pi^{2}}{8}$.
So the answer is: $\frac{\pi^{2}}{8}$.