Derivation of co-channel reuse ratio

The co-channel reuse ratio Q = D/R, where D is the distance between centres of the nearest co-channel cells and R is the cell radius.

Assumptions: hexagonal cells of radius R (centre to vertex); distance between centres of two adjacent cells = √3 R.

Locating a co-channel cell: in a hexagonal layout, the nearest co-channel cell is found by moving i cells along a chain of hexagons, turning 60° anticlockwise, and moving j cells.

                     * Co-channel cell
                    /
                   /  j cells
                  /   (after a 60° turn)
 A *-------------*
      i cells
 D = straight-line distance from A to the co-channel cell

The two paths have lengths i√3R and j√3R, and the angle between them (inside the triangle) is 120°. By the cosine rule:

D² = (i√3R)² + (j√3R)² − 2(i√3R)(j√3R) cos 120°
   = 3R²i² + 3R²j² + 3R²ij        (cos 120° = −1/2)
   = 3R² (i² + ij + j²)

Relating to cluster size: the area of a hexagon is proportional to the square of the distance between centres of neighbouring hexagons. A cluster of N cells repeats with centre spacing D, while single cells repeat with spacing √3R, so

N = (D / √3R)² = i² + ij + j²

Substituting:

D² = 3R² N
D  = R √(3N)
Q  = D / R = √(3N)

A larger Q means more separation between co-channel cells, so less co-channel interference but fewer channels per cell (less capacity).