6. As shown in the figure, in the cube $A B C D-A_{1} B_{1} C_{1} D_{1}$ with edge length 1, $P, Q, R$ are the midpoints of edges $A B, A D, A A_{1}$, respectively. A right triangular prism is constructed with $\triangle P Q R$ as the base, such that the three vertices of its other base are also on the surface of the cube. Then the volume of this right triangular prism is
A. $\frac{3}{8}$
B. $\frac{\sqrt{3}}{8}$
C. $\frac{3}{16}$
D. $\frac{\sqrt{3}}{16}$
Answer
As shown in the figure, the lateral edges of the right triangular prism $P Q R-P_{1} Q_{1} R_{1}$ are parallel to the diagonal $A C_{1}, P_{1}, Q_{1}, R_{1}$ are the midpoints of segments $B C_{1}, D C_{1}, A_{1} C_{1}$, respectively. Given that $S_{\triangle P Q R}=\frac{\sqrt{3}}{4} \cdot\left(\frac{\sqrt{2}}{2}\right)^{2}=\frac{\sqrt{3}}{8}, R R_{1}=\frac{\sqrt{3}}{2}$, thus $V_{P Q R-P_{1} Q_{1} R_{1}}=\frac{\sqrt{3}}{8} \cdot \frac{\sqrt{3}}{2}=\frac{3}{16}$. Therefore, the answer is $C$.