
In
geometry
Geometry (; ) is a branch of mathematics concerned with properties of space such as the distance, shape, size, and relative position of figures. Geometry is, along with arithmetic, one of the oldest branches of mathematics. A mathematician w ...
, Pick's theorem provides a formula for the
area
Area is the measure of a region's size on a surface. The area of a plane region or ''plane area'' refers to the area of a shape or planar lamina, while '' surface area'' refers to the area of an open surface or the boundary of a three-di ...
of a
simple polygon
In geometry, a simple polygon is a polygon that does not Intersection (Euclidean geometry), intersect itself and has no holes. That is, it is a Piecewise linear curve, piecewise-linear Jordan curve consisting of finitely many line segments. The ...
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
Georg Alexander Pick (10 August 1859 – 26 July 1942) was an Austrian Jewish mathematician who was murdered during The Holocaust. He was born in Vienna to Josefa Schleisinger and Adolf Josef Pick and died at Theresienstadt concentration camp. T ...
in 1899. It was popularized in English by
Hugo Steinhaus
Hugo Dyonizy Steinhaus ( , ; 14 January 1887 – 25 February 1972) was a Polish mathematician and educator. Steinhaus obtained his PhD under David Hilbert at Göttingen University in 1911 and later became a professor at the Jan Kazimierz Univers ...
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
be the number of integer points interior to the polygon, and let
be the number of integer points on its boundary (including both vertices and points along the sides). Then the
area
Area is the measure of a region's size on a surface. The area of a plane region or ''plane area'' refers to the area of a shape or planar lamina, while '' surface area'' refers to the area of an open surface or the boundary of a three-di ...
of this polygon is:
The example shown has
interior points and
boundary points, so its area is
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
. 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
, 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
, as needed for the proof. A different proof that these triangles have area
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
, a polygon of area
will be subdivided into
special triangles.
The subdivision of the polygon into triangles forms a
planar graph
In graph theory, a planar graph is a graph (discrete mathematics), graph that can be graph embedding, embedded in the plane (geometry), plane, i.e., it can be drawn on the plane in such a way that its edges intersect only at their endpoints. ...
, and Euler's formula
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
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
, so altogether there are
faces. To count the edges, observe that there are
sides of triangles in the subdivision. Each edge interior to the polygon is the side of two triangles. However, there are
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
, from which one can solve for the number of edges,
. Plugging these values for
,
, and
into Euler's formula
gives
Pick's formula is obtained by solving this
linear equation
In mathematics, a linear equation is an equation that may be put in the form
a_1x_1+\ldots+a_nx_n+b=0, where x_1,\ldots,x_n are the variables (or unknowns), and b,a_1,\ldots,a_n are the coefficients, which are often real numbers. The coeffici ...
for
. An alternative but similar calculation involves proving that the number of edges of the same subdivision is
, 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.
*One can recursively decompose the given polygon into triangles, allowing some triangles of the subdivision to have area larger than 1/2. Both the area and the counts of points used in Pick's formula add together in the same way as each other, so the truth of Pick's formula for general polygons follows from its truth for triangles. Any triangle subdivides its
bounding box
In geometry, the minimum bounding box or smallest bounding box (also known as the minimum enclosing box or smallest enclosing box) for a point set in dimensions is the box with the smallest measure (area, volume, or hypervolume in higher dime ...
into the triangle itself and additional
right triangle
A right triangle or right-angled triangle, sometimes called an orthogonal triangle or rectangular triangle, is a triangle in which two sides are perpendicular, forming a right angle ( turn or 90 degrees).
The side opposite to the right angle i ...
s, and the areas of both the bounding box and the right triangles are easy to compute. Combining these area computations gives Pick's formula for triangles, and combining triangles gives Pick's formula for arbitrary polygons.
*Alternatively, instead of using grid squares centered on the grid points, it is possible to use grid squares having their vertices at the grid points. These grid squares cut the given polygon into pieces, which can be rearranged (by matching up pairs of squares along each edge of the polygon) into a
polyomino
A polyomino is a plane geometric figure formed by joining one or more equal squares edge to edge. It is a polyform whose cells are squares. It may be regarded as a finite subset of the regular square tiling.
Polyominoes have been used in popu ...
with the same area.
*Pick's theorem may also be proved based on
complex integration of a
doubly periodic function related to
Weierstrass elliptic functions.
*Applying the
Poisson summation formula
In mathematics, the Poisson summation formula is an equation that relates the Fourier series coefficients of the periodic summation of a function (mathematics), function to values of the function's continuous Fourier transform. Consequently, the pe ...
to the
characteristic function In mathematics, the term "characteristic function" can refer to any of several distinct concepts:
* The indicator function of a subset, that is the function
\mathbf_A\colon X \to \,
which for a given subset ''A'' of ''X'', has value 1 at points ...
of the polygon leads to another proof.
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
Benchmark may refer to:
Business and economics
* Benchmarking, evaluating performance within organizations
* Benchmark price
* Benchmark (crude oil), oil-specific practices
Science and technology
* Experimental benchmarking, the act of defining a ...
set to test the power of different
proof assistant
In computer science and mathematical logic, a proof assistant or interactive theorem prover is a software tool to assist with the development of formal proofs by human–machine collaboration. This involves some sort of interactive proof edi ...
s. , 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 holes bounded by simple integer polygons, disjoint from each other and from the boundary, has area
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
In mathematics, and more specifically in algebraic topology and polyhedral combinatorics, the Euler characteristic (or Euler number, or Euler–Poincaré characteristic) is a topological invariant, a number that describes a topological space's ...
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
In mathematics, the winding number or winding index of a closed curve in the plane (mathematics), plane around a given point (mathematics), point is an integer representing the total number of times that the curve travels counterclockwise aroun ...
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 polynomial
In mathematics, an integral polytope has an associated Ehrhart polynomial that encodes the relationship between the volume of a polytope and the number of integer points the polytope contains. The theory of Ehrhart polynomials can be seen as a hi ...
s.
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
In mathematics, the Gauss circle problem is the problem of determining how many integer lattice points there are in a circle centered at the origin and with radius r. This number is approximated by the area of the circle, so the real problem is t ...
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
In mathematics, the Farey sequence of order ''n'' is the sequence of completely reduced fractions, either between 0 and 1, or without this restriction, which have denominators less than or equal to ''n'', arranged in order of increasing size.
Wi ...
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
The shoelace formula, also known as Gauss's area formula and the surveyor's formula, is a mathematical algorithm to determine the area of a simple polygon whose vertices are described by their Cartesian coordinates in the plane. It is called the ...
. 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
External links
{{commons category
Pick's Theoremby
Ed Pegg, Jr.
Edward Taylor Pegg Jr. (born December 7, 1963) is an expert on mathematical puzzles and is a self-described recreational mathematician. He wrote an online puzzle column called Ed Pegg Jr.'s ''Math Games'' for the Mathematical Association of Amer ...
, the
Wolfram Demonstrations Project
The Wolfram Demonstrations Project is an Open source, open-source collection of Interactive computing, interactive programmes called Demonstrations. It is hosted by Wolfram Research. At its launch, it contained 1300 demonstrations but has grown t ...
.
Pi using Pick's Theoremby Mark Dabbs,
GeoGebra
Digital geometry
Lattice points
Euclidean plane geometry
Area
Theorems about polygons
Articles containing proofs
Analytic geometry