Based on the diagram shown in the attached file
R = Radius of the sphere
a = Lattice constant
Using pythagoras theorem for the right angled triangle in figure 2
(4R)^2 = a^2 + a^2
16(R^2) = 2(a)^2
Solving the above equation
a=2R√2
To prove that the Atomic Packing factor is 0.74
Let the volume of atom in a unit cell be Vs
Total unit cell volume, Vc
Vc = a^3
Vc =(2R√2)^3
Vc=16R^3 √2 =22.63R^3
Since there are 4 atoms in the FCC unit cell
Vs=4*(4/3 πR^3)
Vs=5.33 πR^3
APF=(5.33 πR^3)/(22.63R^3 )
APF=0.74
