[ [rectangular parallelepipeds]

The diagonals of the faces of a rectangular parallelepiped are $\sqrt{3}, \sqrt{5}$, and 2. Find its volume.
Let $A B C D A 1 B 1 C 1 D 1$ be a rectangular parallelepiped, in which $B D=\sqrt{3}, B C 1=\sqrt{5}, D C 1=2$. Denote $A B=x, A D=y, A A 1=z$. From the right triangles $A B D, B B 1 C 1$ and $C D C 1$ we find that

$$
\left\{\begin{array}{l}
x^{2}+y^{2}=3 \\
x^{2}+z^{2}=5 \\
y^{2}+z^{2}=4
\end{array}\right.
$$

Add the first and second equations of this system term by term and subtract the third from the result. We get $2 \times 2=4$, from which $x=\sqrt{2}$. Similarly, $y=1$ and $z=\sqrt{3}$. Therefore,

$$
V_{A B C D A} 1 B 1 C 1 D 1=x y z=\sqrt{2} \cdot 1 \cdot \sqrt{3}=\sqrt{6} .
$$

## Answer

$\sqrt{6}$.