Enter the interior and boundary lattice-point counts to calculate the ratio of polygon area to one lattice-cell area. This tool is useful for grid-paper geometry when you know the point counts but do not have vertex coordinates.
Use a simple polygon with no holes and with every vertex on the lattice. Its edges must not cross. On square grid paper, lattice points are grid-line intersections.
Count every point strictly inside the polygon in I. Count every distinct point on its edges in B, including vertices and points between vertices. Count each vertex once and exclude boundary points from I. Enter whole numbers only: I must be at least 0 and B must be at least 3.
These conditions follow the setting of UC Davis’s statement of Pick’s theorem.
A rectangle spanning 4 lattice cells across and 3 cells high has 6 interior lattice points and 14 distinct boundary lattice points. Enter I = 6 and B = 14: 6 + 14/2 − 1 = 12. Its area is 12 times one lattice-cell area.
The result uses dimensionless unit 1. A result of 12 means the polygon covers an area equal to 12 lattice cells; it does not mean 12 square metres or another physical area unit. This calculator performs no physical area-unit conversion.
You supply the counts. The tool does not extract points from an image or coordinates, check your counting, or geometrically validate the polygon. A numerical result does not establish that the supplied counts describe a valid polygon. Unavailable arithmetic results are not zero.
If you have vertex coordinates, use the Irregular Polygon Area Calculator, which uses the shoelace method.