A note on area of lattice polygons in an Archimedean tiling

Abstract

Let $$L$$ [3.6.3.6] be the set of vertices generated by the edge-to-edge Archimedean planar tiling using regular triangles and regular hexagons of unit-length, a point $$x\in L$$ [3.6.3.6] be an $$L-$$ point. Suppose the corners of a planar polygon $$P$$ are $$L-$$ points of $$L[3.6.3.6]$$ , then the area of $$P$$ is $$A(P)=\frac{\sqrt{3}}{3}[b(P)+2i(P)+\frac{c(P)}{8}$$ $$-3]-\frac{\sqrt{3}}{12}sc(P)$$ , where $$b(P)$$ is the number of $$L-$$ points on the boundary of $$P, i(P)$$ is the number of $$L-$$ points in the interior of $$P, c(P)$$ is the boundary characteristic of $$P$$ , and $$sc(P)$$ is the side characteristic of $$P$$ .