11.078. A cylinder is formed by rotating a rectangle around one of its sides. Express the volume $V$ of the cylinder in terms of the area $S$ of the rectangle and the length $C$ of the circumference described by the point of intersection of its diagonals.
## Solution.

The volume of the cylinder is found as $V=\pi R^{2} H$ (Fig. 11.70). By construction, $R=a, H=b$. The area of the rectangle $S=a \cdot b=R H$.

![](https://cdn.mathpix.com/cropped/2024_05_22_fa9db84b44b98ec0a5c7g-698.jpg?height=532&width=388&top_left_y=94&top_left_x=190)

Fig. 11.71

![](https://cdn.mathpix.com/cropped/2024_05_22_fa9db84b44b98ec0a5c7g-698.jpg?height=436&width=454&top_left_y=214&top_left_x=729)

Fig. 11.72

The circumference $C=2 \pi r$, where $r=\frac{a}{2}$. Then $C=2 \pi \frac{a}{2}=\pi a=\pi R$. Therefore, we finally get $V=\pi R \cdot R H=C S$.

Answer: $C S$.