Pick Lattice Polygon Area

Pick's theorem provides an elegant formula to determine the area of any lattice polygon (a polygon whose vertices all have integer coordinates) without self-intersections or holes. It relates the exact area of the polygon to the number of integer points inside it and on its boundary.

Worked Example: Primitive Right Triangle

Let's consider a primitive right triangle with vertices at (0, 0), (1, 0), and (0, 1). We can calculate its area using standard geometry and compare it with Pick's theorem.

1. Standard Area Calculation:
   Base = 1, Height = 1
   Area = 0.5 * Base * Height = 0.5

2. Pick's Theorem Variables:
   Vertices: (0, 0), (1, 0), (0, 1)
   Interior lattice points (I) = 0
   Boundary lattice points (B):
     - Edge (0,0) to (1,0): 1 additional point (0,0), length 1
     - Edge (1,0) to (0,1): 1 additional point (1,0), length 1
     - Edge (0,1) to (0,0): 1 additional point (0,1), length 1
   Total B = 3

3. Pick's Formula:
   Area = I + B / 2 - 1
   Area = 0 + 3 / 2 - 1
   Area = 0 + 1.5 - 1 = 0.5

The area calculated by Pick's theorem (0.5) perfectly matches the standard geometric area.

Experiment with the Module

You can interactively verify Pick's theorem using the module controls. Open the Pick Lattice Polygon Area module and try the following steps:

Change the Polygon to "Concave". Observe that the formula still holds perfectly even for non-convex shapes.
Increase the Integer scale to 2 or 3. Notice how the number of interior points (I) scales quadratically, while the boundary points (B) scale linearly, but Pick's theorem dynamically resolves to the correct scaled area.
Adjust the Horizontal shift. The area, interior points, and boundary points remain invariant under translation because the polygon is simply shifted across the lattice.

Limitations and Assumptions

Pick's theorem is remarkably powerful but relies on several strict assumptions:

Lattice Vertices: The formula only works if all vertices of the polygon lie exactly on integer coordinates. It does not apply to polygons with fractional or irrational coordinates.
Simple Polygons: The polygon must be simple, meaning its boundary cannot cross itself (no self-intersections).
Hole-Free: The formula as given applies to solid polygons without holes. If a polygon has h holes, the generalized formula becomes Area = I + B/2 - 1 + h.

References

The following external resources provide mathematical foundations and further generalizations of Pick's Theorem:

Pick Lattice Polygon Area