For the following problems, consider a close$-$packed, three$-$dimensional structure made up of hard spheres all of radius a:

$1$A$)$ Show that the interlayer separation between planes is equal to $1\mathrm{.}633$a

$1$B$)$ Show that the largest sphere that can be inscribed inside the triangle formed by $3$ spheres in the plane of a layer has a radius of $0\mathrm{.}154$a

$1$C$)$ Show that the radius of the tetrahedral holes between the close$-$packed layers is $0\mathrm{.}225$a

$1$D$)$ Show that the radius of the octahedral holes between close$-$packed layers is $0\mathrm{.}414$a

$1$E$)$ Show that the volume fraction of space occupied by the spheres is $0\mathrm{.}741$