4.13. The lateral edges of a triangular pyramid are pairwise perpendicular and have lengths $a, b$ and $c$. Find the volume of the pyramid.
4.13. Let's consider the base of the pyramid to be the right triangle \( A D B \) (Fig. 4.11), and the vertex to be point \( C \). Since \( C D \perp A D \) and \( C D \perp B D \), by the theorem of perpendicularity of a line and a plane, \( C D \perp (A D B) \) and, therefore, the edge \( C D \) is the height of the pyramid. Using formula (2.10), we find the volume of the pyramid:

\[
V=\frac{1}{3} S_{\triangle A B D} \cdot C D=\frac{1}{3} \cdot \frac{1}{2} a b \cdot c=\frac{a b c}{6}
\]

Answer: \(\frac{a b c}{6}\).

![](https://cdn.mathpix.com/cropped/2024_05_21_1e3d8d2f6605ac23e2a5g-579.jpg?height=371&width=342&top_left_y=411&top_left_x=58)

Fig. 4.12