29. Inscribe a cube in a sphere of radius $R$ and find its volume.
29. The diameter of the sphere circumscribed around a cube is equal to the diagonal of the cube: \(2 R = a \sqrt{3}\). Therefore, the edge of the inscribed cube is not difficult to construct (for example, as the radius of the circle circumscribed around an equilateral triangle with sides equal to \(2 R\)). Knowing the length of the edge, we draw a diametrical section of the sphere and inscribe a rectangle with side \(a\) and diagonal \(2 R\) into the resulting circle. We obtain four vertices of the cube. The other four vertices will be obtained in a section perpendicular to the first. The volume of the resulting cube is expressed by the formula:

\[
V=\left(\frac{2 R}{\sqrt{3}}\right)^{3}=\frac{8 R^{3}}{3 \sqrt{3}}
\]