Areas of GeneralizedH-Polygons

LetHbe the set of vertices of a tiling of the plane by regular hexagons of unit area. There is a Pick-type formula which can be applied only to a restricted class of polygons with corners inH. The main purpose of the paper is to give a Pick-type formula applicable to arbitraryH-polygons. The setting is that of general oriented polygons as introduced by Grünbaum and Shephard.