We present a new game, Dots & Polygons, played on a planar point set. Players take turns connecting two points, and when a player closes a (simple) polygon, the player scores its area. We show that deciding whether the game can be won from a given state, is NP-hard. We do so by a reduction from vertex-disjoint cycle packing in cubic planar graphs, including a self-contained reduction from planar 3-Satisfiability to this cycle-packing problem. This also provides a simple proof of the NP-hardness of the related game Dots & Boxes. For points in convex position, we discuss a greedy strategy for Dots & Polygons.