Distance between the centers of two adjacent hexagons in a hexagonal tessellation

Question

Given a hexagonal tessellation where each hexagon has a inradius r, could we say that the distance between two adiacent hexagons is 2r, and in general the distance between any two hexagons is k2r where k is a non negative integer?

Answer 1

Given two hexagons $A$ and $B$ in the teselation, we can find some other $C$ such that $AC$ has $2k$ inradii and $BC$ has $2j$ curcumradii ($k$ or $j$ may be $0$).

Note that the circumradius of the hexagon is $\frac2{\sqrt 3}r$ where $r$ is the inradius.

Since the angle between $AC$ and $BC$ is known ($120$ degrees) we can apply the Law of the Cosine:

$$AB^2=4k^2r^2+\frac{16}3j^2r^2-\frac{16}{\sqrt 3}jkr^2\cos120^o$$

After some algebra work, this gives $$AB=2r\sqrt{(j+k)^2+\frac{j^2}3}$$