Pick's Theorem

Lattice-polygon area: A = i + b/2 - 1.

The idea

Theorem (Pick's theorem).

Let a polygon have all its vertices at lattice points — points whose coordinates are both integers — and a boundary that does not cross itself. If $i$ counts the lattice points strictly inside it and $b$ those on its boundary, then its area is $A = i + \frac{b}{2} - 1.$

Pick's theorem computes the area of a polygon drawn on a grid by counting points. The formula uses no side lengths, no angles, and no coordinates — only the two counts, whatever the number of sides.

To see what each term measures, give every lattice point the unit square centered on it and ask how much of that square lies inside the polygon. An interior point contributes its whole square. A boundary point contributes about half, a little more or less where the boundary turns a corner; summed around any closed boundary, those corner discrepancies total exactly one square, and the $-1$ subtracts it.

Ways to work on it

• Walkthrough. Area of a lattice square from its interior and boundary point counts.
• Practice. Compute area from randomized interior and boundary point counts.
• Hardest. Recover the interior count from the area, then count boundary points on slanted edges.

Not sure where to start? Take the ten-question placement test.