Lattice Polgyons Part 2

In the previous post, I introduced the idea of a lattice polygon and a method for calculating the area. I want to discuss a very powerful theorem.

Before we can state the theorem, we need to identify two types of points.

  • A boundary point (B) is a point that lies on one of the lines of the lattice polygon
  • An interior point (I) is a point that is contained inside the lattice polygon

In the lattice polygon below the boundary points are in blue and the interior points are in red.

polygon 1

This lattice polygon has B = 11 (blue points) and I = 6 (red points)

The theorem, formulated in 1899 by Georg Pick, states the following:

Area = \frac{1}{2}*B+I-1

Therefore, the theorem states that our previous lattice polygon has an area of:

Area = \frac{1}{2}*11+6-1 = 10.5

You can use the method from the previous post and verify that the area is indeed 10.5. This theorem is cool because you can take any huge and complicated lattice polygon and find the area with a super simple formula. We have no need to resort to cutting up the shape into smaller triangles and rectangles. For example, find the area of the following lattice polygon:

polygon 2

You should find B = 25, I = 10, and area = 21.5. Pretty cool.

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