For a simple polygon in the plane whose vertices lie on integer lattice points, Pick's theorem expresses the Euclidean area A as A = I + B/2 − 1, where I is the number of interior lattice points and B is the number of lattice points on the boundary.

Principle

Area of a lattice polygon is combinatorially determined by counting lattice points in the interior and on the boundary; geometry reduces to lattice counting for integer-coordinate vertices.

Demonstration

Example: triangle with vertices at (0,0), (4,0) and (0,3). Its area is 6 by the shoelace formula. Boundary lattice points B are 8 (five on the base, four on the vertical side, two on the hypotenuse counted with endpoints and adjusted), so Pick predicts I = A − B/2 + 1 = 6 − 4 + 1 = 3 interior lattice points, which can be checked directly.

Misapplication

Applying the formula to polygons with non-integer vertices, to polygons with holes, or to non-simple (self-intersecting) polygons; Pick's theorem fails or needs modification in these cases.

Consequence

Gives an exact, simple combinatorial method for computing areas of lattice polygons and for deducing interior lattice counts from area and boundary data; it underlies discrete and computational approaches to planar lattice geometry.

Reversal

Given A and B for a lattice polygon, Pick's theorem yields I uniquely; however, area data alone do not determine combinatorial structure of the polygon. The formula is an equality rather than an implication about shape beyond lattice counts.

Boundary

Requires a simple (non-self-intersecting) polygon with vertices at integer-coordinate lattice points in the plane; polygons with holes, curved boundaries, or non-integer vertices fall outside scope.

Semantic Tension

Often seen in relation to discrete versions of continuous theorems (for instance Gauss-Bonnet or Euler characteristic analogues); tension arises between combinatorial lattice-count statements and smooth-area concepts but Pick's theorem is exact in the lattice setting.

Synthesis

Pick's theorem provides an exact bridge between discrete lattice counts and continuous area for simple lattice polygons: area equals interior points plus half the boundary points minus one, offering a compact combinatorial formula for planar lattice geometry.