Since the shaft is hollow, it will have both an outside diameter and an inside diameter. To distinguish between the two, the symbol $\mathrm{D}$ will be used for the outside diameter and the symbol $\mathrm{d}$ will be used for the inside diameter.

For a hollow shaft, the polar moment of inertia $\mathrm{J}$ is defined as:

$\mathrm{J}=\frac{\mathrm{\pi }}{32}\left({\mathrm{D}}^4-{\mathrm{d}}^4\right)$

where $\mathrm{D}$ is the outside diameter and $\mathrm{d}$ is the inside diameter. Factor out the term ${\mathrm{D}}^4$ and simplify to obtain:

$\mathrm{J}=\frac{\mathrm{\pi }}{32}{\mathrm{D}}^4\left[1-\frac{{\mathrm{d}}^4}{{\mathrm{D}}^4}\right]=\frac{\mathrm{\pi }}{32}{\mathrm{D}}^4\left[1-{\left(\frac{\mathrm{d}}{\mathrm{D}}\right)}^4\right]$

where $(\mathrm{d}/\mathrm{D})$ is the ratio of inside diameter to outside diameter.