proof of Pick’s theorem

Pick’s theorem:

Let P2 be a polygonMathworldPlanetmathPlanetmath with all vertices on lattice points on the grid 2. Let I be the number of lattice points that lie inside P, and let O be the number of lattice points that lie on the boundary of P. Then the area of P is


To prove, we shall first show that Pick’s theorem has an additive character. Suppose our polygon has more than 3 vertices. Then we can divide the polygon P into 2 polygons P1 and P2 such that their interiors do not meet. Both have fewer vertices than P. We claim that the validity of Pick’s theorem for P is equivalentMathworldPlanetmathPlanetmathPlanetmathPlanetmathPlanetmath to the validity of Pick’s theorem for P1 and P2.

Denote the area, number of interior lattice points and number of boundary lattice points for Pk by Ak,Ik and Ok, respectively, for k=1,2.

Clearly A=A1+A2.

Also, if we denote the number of lattice points on the edges common to P1 and P2 by L, then






This proves the claim. Therefore we can triangulate P and it suffices to prove Pick’s theorem for trianglesMathworldPlanetmath. Moreover by further triangulations we may assume that there are no lattice points on the boundary of the triangle other than the vertices. To prove pick’s theorem for such triangles, embed them into rectanglesMathworldPlanetmathPlanetmath.

Again by additivity, it suffices to prove Pick’s theorem for rectangles and rectangular triangles which have no lattice points on the hypotenuseMathworldPlanetmath and whose other two sides are parallelMathworldPlanetmathPlanetmath to the coordinate axes. If these two sides have lengths a and b, respectively, we have




Furthermore, by thinking of the triangle as half of a rectangle, we get


(Note that here it is essential that no lattice points are on the hypotenuse) From these equations for A,I and O, Pick’s theorem is satisfied for these triangles.

Finally for a rectangle, whose sides have lengths a and b, we find that




From these Pick’s theorem follows for rectangles too. This completesPlanetmathPlanetmathPlanetmathPlanetmath our proof.

Title proof of Pick’s theorem
Canonical name ProofOfPicksTheorem
Date of creation 2013-03-22 13:09:47
Last modified on 2013-03-22 13:09:47
Owner giri (919)
Last modified by giri (919)
Numerical id 5
Author giri (919)
Entry type Proof
Classification msc 51A99
Classification msc 05B99
Classification msc 68U05