11.004. Determine the volume of an oblique triangular prism, where the area of one of the lateral faces is $S$, and the distance from the plane of this face to the opposite edge is $d$.
## Solution.

Through point $K$ on edge $A A^{\prime}=l$ (Fig. 11.6), we draw a section perpendicular to this edge. Then $K M \perp B B^{\prime}$, since $B B^{\prime} \| A A^{\prime} ; K N \perp C C^{\prime}$, since $C C^{\prime} \| A A^{\prime}$. In $\triangle K M N$, we draw the altitude $K D . K D \perp M N$ and $K D \perp C C^{\prime}$, therefore, $K D \perp B B^{\prime} C^{\prime} C$ and $K D=d$. By the condition, the area of face $B B^{\prime} C^{\prime} C$ is $S$. Then $V=S_{\triangle K M N} \cdot A A^{\prime}=\frac{1}{2} K D \cdot M N \cdot A A^{\prime}=\frac{1}{2} d \cdot a \cdot l$, where $M N=a, M N \perp C C^{\prime}=l$.

Further, $V=\frac{1}{2} d \cdot(a \cdot l)=\frac{1}{2} d\left(M N \cdot C C^{\prime}\right)=\frac{1}{2} d S$.

Answer: $V=\frac{1}{2} d S$.