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 $P$.