Let's calculate the volume of a cube where the diagonal of one face is $3 \mathrm{~cm}$. What is the diagonal of the cube?
If the edge of a cube is $a$, and the diagonal of one of its faces is $d$, then

$$
d^{2}=a^{2}+a^{2}=2 a^{2}
$$

thus

$$
a=\frac{d}{\sqrt{2}}=\frac{d \sqrt{2}}{2}
$$

The volume of the cube is

$$
K=a^{3}=\frac{d^{3}}{4} \sqrt{2}=9.545 \mathrm{~cm}^{3}
$$

and its space diagonal is

$$
s=\sqrt{d^{2}+a^{2}}=\sqrt{3 a^{2}}=a \sqrt{3}=\frac{d \sqrt{6}}{2}=3.67 \mathrm{~cm}
$$

(Lajos Königs, Késmárk.)