Unlocking Lattice Geometry: How Pick’s Theorem Calculates Polygon Areas

1. Introduction to Pick’s Theorem

Pick’s Theorem provides a method to calculate the area of a simple (non‑self‑intersecting) polygon whose vertices lie on integer lattice points. It was first stated by Georg Pick in 1899.

2. Formula

For such a polygon, let B be the number of lattice points on its boundary and I the number of interior lattice points. The area A is given by:

A = I + B/2 - 1

3. Sketch of Proof

The proof typically decomposes the polygon into triangles, computes each triangle’s area using the formula, and sums them, showing the relationship holds for the whole polygon.

4. Example Application

Consider a rectangle with opposite vertices at (0,0) and (4,3). Counting lattice points:

Boundary points: 5 on the bottom edge, 4 on the right edge, 5 on the top edge, and 4 on the left edge, giving B = 14 after removing the double‑counted corners.

Interior points: a 3 × 2 grid yields I = 6 interior points.

Applying Pick’s Theorem:

A = 6 + 14/2 - 1 = 12

This matches the rectangle’s geometric area (width × height = 4 × 3 = 12).

Pick’s Theorem thus offers a simple, intuitive way to compute areas of lattice polygons, especially when traditional formulas or integration are inconvenient.