6. Inscribed in a cube with edge a is a right circular cylinder such that the axis of the cylinder contains the diagonal of the cube, and the circles of the cylinder's bases touch the faces of the cube. Calculate the volume of this cylinder.
6. Place a right circular cylinder so that its axis coincides with the diagonal of a cube, and the center of the cube is equidistant from the bases of the cylinder. Draw a diagonal section through this diagonal of the cube, and we will obtain the figure shown in Figure 105. Take the center of the cube as the center of homothety that maps point $A^{\prime}$ to $A$; under this homothety, our cylinder will transform into a cylinder inscribed in the cube. To find the radius of the cylinder $r$, consider the similar triangles $A M Q$ and $M N P$. We get:

$$
\frac{|M Q|}{|P N|}=\frac{|A Q|}{|M P|}
$$

here $|M Q|=\frac{a \sqrt{3}}{2}-r,|P N|=a \sqrt{2},|A Q|=r,|M P|=a$. From equality (1), we find: $\frac{\frac{a \sqrt{3}}{2}-r}{a \sqrt{2}}=\frac{r}{a}, \quad a \sqrt{3}=2 r(1+\sqrt{2})$, $r=\frac{a \sqrt{3}}{2(1+\sqrt{2})}$. Knowing the radius, we can calculate the lateral surface area and the volume of the cylinder using the formulas (see, for example, the solution to the previous problem).