##  [Pick's Theorem](/picks-theorem-0) 

 Definition

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

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

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

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

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

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

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

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

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.