11.145. The lateral faces of a triangular pyramid are mutually perpendicular, and their areas are equal to $a^{2}, b^{2}$ and $c^{2}$. Determine the volume of the pyramid.
## Solution.

Let $S A=x, S B=y, S C=z$ (Fig. 11.42). From the mutual perpendicularity of the lateral faces, it follows that these edges are mutually perpendicular, hence the desired volume $V=\frac{1}{3} S_{\triangle A S B} \cdot S C=\frac{1}{6} x y z$. By the given conditions $\frac{1}{2} x y=a^{2}, \frac{1}{2} x z=b^{2}, \frac{1}{2} y z=c^{2}$. Multiplying these equations, we get $\frac{1}{8} x^{2} y^{2} z^{2}=a^{2} b^{2} c^{2}$. Therefore, $V=\frac{1}{6} a b c \sqrt{8}=\frac{1}{3} a b c \sqrt{2}$.

Answer: $\frac{1}{3} a b c \sqrt{2}$.