GeoProof — Pick's Theorem explorer

interior point (I) boundary point (B) other grid point

Area = I + B/2 − 1

Pick's Theorem

For a simple polygon whose vertices all land on grid (lattice) points, the area is found just by counting dots:

Area = I + B/2 − 1

I = grid points strictly inside, B = grid points on the boundary (including the corners). No lengths or coordinates needed.

A quick sense of it

Each interior point "owns" a full unit of area; each boundary point owns about half; and the −1 is a fixed correction for going once around a closed loop. Remarkably, it's exact for every lattice polygon.

Counting the boundary

An edge from one lattice point to another passes through gcd(|Δx|, |Δy|) lattice steps, so the number of boundary points is the sum of those gcds over all edges.

Keep it simple

Pick's theorem needs a simple polygon — edges that don't cross. If you drag the corners so the shape self-intersects, the count no longer matches the area; the badge will warn you.