In Diamond Cubic Packing, we have:
4 atoms at tetrahedral voids with its contribution of 1 atom per unit cell
(1 x 4 = 4),
6 atoms at Face center with its contribution as 1/2 per unit cell
(1/2 x 6 = 3),
8 atoms at Corners with its contribution as 1/8 per unit cell (1/8 x 8 = 1).

So, number of atoms per unit cell (n)= 4 + 3 + 1 = 8
Now,

Atomic Packing Factor (A.P.F) =
{n x (Total volume of atom in a unit cell)}/{Volume of unit cell}
= {8 x (4/3) x 3.141 x (R^3)}/ a^3

Here, a (Edge Length) = 8R/(3)^1/2 = 8R/root(3).
= ((3)^1/2) x 3.141 / 16 = 0.34

Therefore,
APF = 0.34
Packing efficiency will be (0.34 x 100) = 34%