A primitive cell is a unit cell that contains exactly one lattice point.  For unit cells generally, lattice points that are shared by  cells are counted as  of the lattice points contained in each of those cells; so for example a primitive unit cell in three dimensions which has lattice points only at its eight vertices is considered to contain  of each of them. An alternative conceptualization is to consistently pick only one of the  lattice points to belong to the given unit cell (so the other  lattice points belong to adjacent unit cells).
The primitive translation vectors , ,  span a lattice cell of smallest volume for a particular three-dimensional lattice, and are used to define a crystal translation vector
where , ,  are integers, translation by which leaves the lattice invariant. That is, for a point in the lattice , the arrangement of points appears the same from  as from .
Since the primitive cell is defined by the primitive axes (vectors) , , , the volume  of the primitive cell is given by the parallelepiped from the above axes as
Usually, primitive cells in two and three dimensions are chosen to take the shape parallelograms and parallelepipeds, with an atom at each corner of the cell. This choice of primitive cell is not unique, but volume of primitive cells will always be given by the expression above.