How do you find the packing efficiency for hexagonal closest packed crystal lattice?

Question

Can you tell me what the sides are? and what volumes to use? Thanks

Answer 1

working . . .

Extend your time on your question if you can, please.

Edit:

Assume the lattice is made up of spheres, all of which have the radius, R.

Envision a rectangular solid with dimensions 2R, 2R√3, and 2R√3.  These are the center-to-center distances of the spheres at the 8 vertices of the solid.  This volume will contain the volume equivalent of 5 spheres. The total volume of the rectangular solid is

Vr = (2R)(2R√3)(2R√3) = 24R^3

The volume of the part-spheres enclosed is

Vs = (5)(4/3)πR^3 = (20/3)πR^3

The packing efficiency is the ratio

Vs/Vr = ((20/3)πR^3) / (24R^3)

Vs/Vr = 5π/18

Vs/Vr ≈ 0.8726646 ≈ 87.3%