Lattice Points & Pick's Theorem Cheat Sheet

Key Facts

• A lattice point is a point $(x,y)$ where both $x$ and $y$ are integers.
• Pick's Theorem states that the area of a simple lattice polygon is $A = I + \frac{B}{2} - 1$.
• In Pick's Theorem, $I$ is the number of lattice points strictly inside the polygon and $B$ is the number of lattice points on the boundary.
• For a segment from $(x_1,y_1)$ to $(x_2,y_2)$, the number of boundary lattice points on the segment including both endpoints is $\gcd(|x_2-x_1|,|y_2-y_1|)+1$.
• When adding boundary counts around a polygon, use $B = \sum \gcd(|\Delta x|,|\Delta y|)$ so vertices are not double-counted.
• Pick's Theorem can be rearranged to find interior points: $I = A - \frac{B}{2} + 1$.
• Pick's Theorem can be rearranged to find boundary points: $B = 2A - 2I + 2$.
• Pick's Theorem applies only to simple polygons whose vertices are lattice points, not to curves, self-intersecting polygons, or polygons with non-lattice vertices.

Vocabulary

Lattice point

A point on the coordinate plane with integer coordinates, such as $(3,-2)$.

Lattice polygon

A polygon whose vertices are all lattice points.

Boundary point

A lattice point that lies on an edge or vertex of the polygon.

Interior point

A lattice point that lies strictly inside the polygon and not on any edge.

Pick's Theorem

A formula for the area of a simple lattice polygon, given by $A = I + \frac{B}{2} - 1$.

Greatest common divisor

The greatest common divisor, written $\gcd(a,b)$, is the largest positive integer that divides both $a$ and $b$.