Pick's theorem

Pick's theorem states that a simple polygon with lattice-point vertices has area I + B/2 − 1, where I and B count its interior and boundary lattice points.

Lattice point: A point in Euclidean space whose coordinates are integers in a chosen coordinate system. These integer-coordinate points are the units counted by both I and B.
Triangulation: A subdivision of a polygon or other region into triangles using noncrossing diagonals. Decomposing a lattice polygon into primitive triangles gives a route to the formula.
Lattice polygon area computation: The calculation of a lattice polygon's area from its interior and boundary lattice-point counts. This is the theorem's direct use when point counts are easier than geometric measurement.
Green's theorem: A theorem relating a line integral around a planar region to a double integral over that region. It obtains area from boundary data through calculus rather than lattice-point counts.
Lattice polygon: A polygon whose vertices lie on the integer lattice. Pick's theorem applies to this class of polygons.
Unimodular triangle: A lattice triangle whose area is one half, equivalently whose edge vectors have determinant of absolute value one. Every lattice triangle can be subdivided into these minimal-area pieces.
Integer programming: Optimization in which decision variables are restricted to integer values. Lattice-point counts connect polygon geometry with feasible integer solutions.
Cavalieri's principle: A method for comparing areas or volumes through equal cross-sectional measures. It derives area by comparing slices, without requiring vertices on an integer lattice.
Simple polygon: A polygonal curve that does not intersect itself except at consecutive shared endpoints. The theorem assumes a non-self-intersecting polygonal boundary.
Shoelace formula: A coordinate formula for the area of a polygon whose vertices are listed in order. It computes the same area directly from the polygon's lattice coordinates.

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QuadrilateralRelated: For quadrilaterals with lattice-point vertices, it offers an alternative area calculation.