10. A rectangle with length $\sqrt{2}$ and width 1, when rotated one full turn about the line containing one of its diagonals, find the volume of the resulting solid of revolution.
Given $A B=\sqrt{2}, A D=1$, the rectangle $A B C D$ rotates around $A C$ for one full turn, resulting in the solid of revolution as shown in the right figure. Let the volume of the solid be $V$. Then $D E=\frac{\sqrt{2}}{\sqrt{3}}=\frac{\sqrt{6}}{3}, A E=\frac{1}{\sqrt{3}}=\frac{\sqrt{3}}{3}, O E=\frac{\sqrt{3}}{6}$, thus $\frac{O N}{D E}=\frac{3}{4} \Rightarrow O N=\frac{\sqrt{6}}{4}$.
Therefore, $\frac{1}{2} V=V_{\text {cone } A E}+V_{\text {cone } C E}-V_{\text {cone } C O}$ $=\frac{1}{3} \pi\left[\left(\frac{\sqrt{6}}{3}\right)^{2} \cdot \sqrt{3}-\left(\frac{\sqrt{6}}{4}\right)^{2} \cdot \frac{\sqrt{3}}{2}\right]=\frac{23 \sqrt{3} \pi}{144} \Rightarrow V=\frac{23 \sqrt{3} \pi}{72}$.