What is the relation between edge length $\mathrm{a}$ and radius of an atom $\mathrm{r}$ in $\mathrm{HCP}$ unit cell?

The edge length of a solid possessing cubic unit cell is $2\sqrt{2}\mathrm{r}$ (structure I), based on hard sphere model, which upon subjecting to a phase transition, a new cubic structure (structure II) having an edge length of $\frac{4\mathrm{r}}{\sqrt{3}}$ is obtained, where $\mathrm{r}$ is the radius of the hard sphere. Which of the following statements is true ?

A metal $\mathrm{M}$ forms hexagonal close-packed structure. The total number of voids in $0.02 \mathrm{mol}$ of it is _____ $\times {10}^{21}$ (Nearest integer)

(Given ${\mathrm{N}}_{\mathrm{A}}=6.02\times {10}^{23}$)

In a face centered cubic (fcc) lattice, the position which represents an octahedral hole is

A two dimensional solid is made by alternating circles with radius $\mathrm{a}$ and $\mathrm{b}$ such that the sides of the circles touch. The packing fraction is defined as the ratio of the area under the circles to the area under the rectangle with sides of length $\mathrm{x}$ and $\mathrm{y}.$

The ratio $\mathrm{r}=\mathrm{b}/\mathrm{a}$ for which the packing fraction is minimized is closest to:

$\mathrm{Zn}$ converts from molten state to its solid state in $\mathrm{hcp}$ structure. What is the number of the nearest atoms?