Formula for area of a grid polygon
In geometry, Pick's theorem provides a formula for the area of a simple polygon with integer vertex coordinates, in terms of the number of integer points within it and on its boundary. The result was first described by Georg Alexander Pick in 1899. It was popularized in English by Hugo Steinhaus in the 1950 edition of his book "Mathematical Snapshots". It has multiple proofs, and can be generalized to formulas for certain kinds of non-simple polygons.
Formula.
Suppose that a polygon has integer coordinates for all of its vertices. Let formula_1 be the number of integer points interior to the polygon, and let formula_2 be the number of integer points on its boundary (including both vertices and points along the sides). Then the area formula_3 of this polygon is:
formula_4
The example shown has formula_5 interior points and formula_6 boundary points, so its area is formula_7 square units.
Proofs.
Via Euler's formula.
One proof of this theorem involves subdividing the polygon into triangles with three integer vertices and no other integer points. One can then prove that each subdivided triangle has area exactly formula_8. Therefore, the area of the whole polygon equals half the number of triangles in the subdivision. After relating area to the number of triangles in this way, the proof concludes by using Euler's polyhedral formula to relate the number of triangles to the number of grid points in the polygon.
The first part of this proof shows that a triangle with three integer vertices and no other integer points has area exactly formula_8, as Pick's formula states. The proof uses the fact that all triangles tile the plane, with adjacent triangles rotated by 180° from each other around their shared edge. For tilings by a triangle with three integer vertices and no other integer points, each point of the integer grid is a vertex of six tiles. Because the number of triangles per grid point (six) is twice the number of grid points per triangle (three), the triangles are twice as dense in the plane as the grid points. Any scaled region of the plane contains twice as many triangles (in the limit as the scale factor goes to infinity) as the number of grid points it contains. Therefore, each triangle has area formula_8, as needed for the proof. A different proof that these triangles have area formula_8 is based on the use of Minkowski's theorem on lattice points in symmetric convex sets.
This already proves Pick's formula for a polygon that is one of these special triangles. Any other polygon can be subdivided into special triangles: add non-crossing line segments within the polygon between pairs of grid points until no more line segments can be added. The only polygons that cannot be subdivided in this way are the special triangles considered above; therefore, only special triangles can appear in the resulting subdivision. Because each special triangle has area formula_8, a polygon of area formula_3 will be subdivided into formula_14 special triangles.
The subdivision of the polygon into triangles forms a planar graph, and Euler's formula formula_15 gives an equation that applies to the number of vertices, edges, and faces of any planar graph. The vertices are just the grid points of the polygon; there are formula_16 of them. The faces are the triangles of the subdivision, and the single region of the plane outside of the polygon. The number of triangles is formula_14, so altogether there are formula_18 faces. To count the edges, observe that there are formula_19 sides of triangles in the subdivision. Each edge interior to the polygon is the side of two triangles. However, there are formula_2 edges of triangles that lie along the polygon's boundary and form part of only one triangle. Therefore, the number of sides of triangles obeys the equation formula_21, from which one can solve for the number of edges, formula_22. Plugging these values for formula_23, formula_24, and formula_25 into Euler's formula formula_15 gives
formula_27
Pick's formula is obtained by solving this linear equation for formula_3. An alternative but similar calculation involves proving that the number of edges of the same subdivision is formula_29, leading to the same result.
It is also possible to go the other direction, using Pick's theorem (proved in a different way) as the basis for a proof of Euler's formula.
Other proofs.
Alternative proofs of Pick's theorem that do not use Euler's formula include the following.
Pick's theorem was included in a 1999 web listing of the "top 100 mathematical theorems", which later became used by Freek Wiedijk as a benchmark set to test the power of different proof assistants. As of 2024[ [update]], Pick's theorem had been formalized and proven in only two of the ten proof assistants recorded by Wiedijk.
Generalizations.
Generalizations to Pick's theorem to non-simple polygons are more complicated and require more information than just the number of interior and boundary vertices. For instance, a polygon with h holes bounded by simple integer polygons, disjoint from each other and from the boundary, has area
formula_30
It is also possible to generalize Pick's theorem to regions bounded by more complex planar straight-line graphs with integer vertex coordinates, using additional terms defined using the Euler characteristic of the region and its boundary, or to polygons with a single boundary polygon that can cross itself, using a formula involving the winding number of the polygon around each integer point as well as its total winding number.
The Reeve tetrahedra in three dimensions have four integer points as vertices and contain no other integer points, but do not all have the same volume. Therefore, there does not exist an analogue of Pick's theorem in three dimensions that expresses the volume of a polyhedron as a function only of its numbers of interior and boundary points. However, these volumes can instead be expressed using Ehrhart polynomials.
Related topics.
Several other mathematical topics relate the areas of regions to the numbers of grid points. Blichfeldt's theorem states that every shape can be translated to contain at least its area in grid points. The Gauss circle problem concerns bounding the error between the areas and numbers of grid points in circles. The problem of counting integer points in convex polyhedra arises in several areas of mathematics and computer science.
In application areas, the dot planimeter is a transparency-based device for estimating the area of a shape by counting the grid points that it contains. The Farey sequence is an ordered sequence of rational numbers with bounded denominators whose analysis involves Pick's theorem.
Another simple method for calculating the area of a polygon is the shoelace formula. It gives the area of any simple polygon as a sum of terms computed from the coordinates of consecutive pairs of its vertices. Unlike Pick's theorem, the shoelace formula does not require the vertices to have integer coordinates.
References.
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