b
\(\begin{array}{l}
Here\,a = \frac{2}{{\sqrt 3 }}R\\
Now,\frac{M}{{M'}} = \frac{{\frac{4}{3}\pi {R^3}}}{{{a^3}}}\\
 = \frac{{\frac{4}{3}\pi {R^3}}}{{{{\left( {\frac{2}{{\sqrt 3 }}R} \right)}^3}}} = \frac{{\sqrt 3 }}{2}\pi .\,M' = \frac{{2M}}{{\sqrt 3 \pi }}
\end{array}\)

Moment of inertia of the cube about the given axis,

\(I = \frac{{M'{a^2}}}{6} = \frac{{\frac{{2M}}{{\sqrt 3 \pi }} \times {{\left( {\frac{2}{{\sqrt R }}R} \right)}^2}}}{6} = \frac{{4M{R^2}}}{{9\sqrt 3 \pi }}\)