An Elementary Proof of Pick's Theorem

Abstract

Pick's Theorem, a statement of the relationship between the area of a polygonal region on a lattice and its interior and boundary lattice points, is familiar to those whose students have participated in activities and discovery lessons using the geoboard. This theorem provides an opportunity for mathematics students to use their previous knowledge and powers of deductive reasoning to establish a result that is rather surprising. Although proofs of Pick's Theorem are not unknown (Coxeter, 1969;DeTemple and Robertson, 1974), they depend on a mathematical sophistication which is beyond the grasp of the average geometry student. The proof presented in this paper requires only intuitions, relationships, and skills established prior to the student's taking a one year course in geometry. Although the proof is rather long, the reasoning required is not difficult.Pick's Theorem: If P is a polygonal region whose vertices are lattice points, and P has B lattice points on its boundary and I lattice points in its interior, the area of P is given by A = B/2 + I -1. This result often comes as the final generalization of a guided discovery lesson with the geoboard (Fitzgerald et al., 1973;Hirsch, 1974). Although the result is rather easy to establish, its proof is more elusive. First we will show that the result is true for a special type of triangular region. We will then generalize the result to any polygonal region whose vertices are lattice points. Although the results to follow can be established for an arbitrary lattice, the lattice used here is the set of points in the Cartesian plane with integral coordinates. Thus, the geoboard is an ideal model of this lattice. SPECIAL CASE Consider any triangle which can be constructed on the lattice with no boundary points other than its three vertices. A triangle of this type will be called an elementary triangle. An elementary triangle is illustrated in figure 1. Without loss of generality we may consider this elementary triangle to be in standard position, that is, entirely within the first quadrant and with one of its vertices at the origin. See figure 2. A little experimentation with the geoboard will serve to convince the student that this placement is always possible. We may now find the area of the triangular region by enclosing it in a rectangular region and subtracting the areas of the three triangular regions formed from that of the rectangular region.

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