lattice points per unit cell
How many lattice points belong to one unit cell? You cannot just count every dot you see, because points on corners, edges, and faces are shared with the neighbouring cells that press up against them. You count each shared point by the fraction of it that lies inside your own cell.
The sharing rules in 3D: a corner point is shared by 8 cells (counts 1/8), an edge point by 4 (counts 1/4), a face point by 2 (counts 1/2), and a point fully inside counts 1. Worked examples: primitive P = 8 times 1/8 = 1; body-centred I = 8 times 1/8 + 1 = 2; face-centred F = 8 times 1/8 + 6 times 1/2 = 4; base-centred C = 8 times 1/8 + 2 times 1/2 = 2.
This count equals the ratio of the conventional cell volume to the primitive cell volume, and it feeds straight into the theoretical density: multiply lattice points per cell by atoms per motif to get atoms per cell, then density = (atoms per cell times atomic mass) / (cell volume times Avogadro's number). Miscounting shared atoms is the single most common arithmetic mistake beginners make.
Count the FCC cube: 8 corner points shared 8 ways give 8 times 1/8 = 1, and 6 face points shared 2 ways give 6 times 1/2 = 3, total 4. With one atom per lattice point, that is the well-known 4 atoms per FCC unit cell.
8 corners (divided by 8) + 6 faces (divided by 2) = 1 + 3 = 4 points in the FCC cell.
The sharing fractions (1/8 corner, 1/4 edge, 1/2 face) assume a parallelepiped cell that tiles space face-to-face; they are a bookkeeping trick, not physics, and must be re-derived for oddly shaped cells such as the hexagonal prism.