Why not a rhombus, with increasingly acute angles for increasingly smaller area?
Yes, that would produce a 12-perimeter polygon with a tiny area (approaching 0). But in this problem we are restricted to integers only.
The toothpick is a simple instrument with a simple purpose. We might not give it a second thought as we pick one up, free our teeth from discomfort and unsightly food remains, and then casually toss it away. Cinnamon Toothpicks