$(20$ points$)$ You are evaluating a variety of metallic materials to use them in the femoral stem of a hip joint replacement device. Diagrams of the crystal structures of the material used are shown in $($i$),($ii$)$ and $($iii$).$

A$.$ Identify the type of each crystal structure and their coordination number. $(2$ points$)$

B$.$ Derive the relation between $r($radius of sphere$)$ and $a($length of cube side$)$ of each crystal structure. Show your works $(8$ points$)$

C$.$ Determine the atomic packing factor $($APF$)$ of each crystal structures. Show your works $(6$ points$)$