Pick Lattice Polygon Area
Count dots inside and on the boundary of a lattice polygon to recover its area exactly.
Free theory articleCount dots inside and on the boundary of a lattice polygon to recover its area exactly.
Concept & interpretation
Pick's theorem provides an elegant formula to determine the area of any lattice polygon (a polygon whose vertices all have integer coordinates) without self-intersections or holes. It relates the exact area of the polygon to the number of integer points inside it and on its boundary.
Experiment with the Module
You can interactively verify Pick's theorem using the module controls. Open the Pick Lattice Polygon Area module and try the following steps:
- Change the Polygon to "Concave". Observe that the formula still holds perfectly even for non-convex shapes.
- Increase the Integer scale to 2 or 3. Notice how the number of interior points (I) scales quadratically, while the boundary points (B) scale linearly, but Pick's theorem dynamically resolves to the correct scaled area.
- Adjust the Horizontal shift. The area, interior points, and boundary points remain invariant under translation because the polygon is simply shifted across the lattice.