## Task 11 - V00911

A sphere with radius $r_{u}=1$ is to be inscribed with a cube. How long will the edge $a$ of the cube be? The cube is to be inscribed with another sphere. How long will the radius $r_{i}$ of this sphere be?
The sphere is the so-called circumsphere of the cube, where the cube points lie on the sphere. Thus, the radius $r_{u}$ is equal to half the length of the space diagonal of the cube, i.e.,

$$
\frac{\sqrt{3 a^{2}}}{2}=r_{u}=1 \quad \Rightarrow \quad a=\frac{2}{3} \sqrt{3}
$$

The diameter of the insphere is equal to the edge length of the cube, i.e.,

$$
r_{i}=\frac{a}{2}=\frac{\frac{2}{3} \sqrt{3}}{2}=\frac{1}{3} \sqrt{3}
$$