Pick's Theorem
Pick's Theorem expresses the area of a polygon, all of whose vertices are lattice points in a coordinate plane, in terms of the number of lattice points inside the polygon and the number of lattice points on the sides of the polygon. The formula is:
where is the number of lattice points in the interior and
being the number of lattice points on the boundary.
It is similar to the Shoelace Theorem, and although it is less powerful, it is a good tool to have in solving problems.
Proof
Iff a triangle on the lattice points with no point in its interior or on its edges, it has an area of . Such triangle must contain two lattice points distance
from each other and one lattice point on a line parallel to the opposite edge distance
apart. The minimum distance between two distinct lattice points is
. If no two lattice points have distance
, by
the area is more than 1 and similarly for the height.
Removing
of the mentioned triangles either removes
boundary point, turns
interior point into a boundary point, accounting for the
part. The
part is accounted for by looking at the area of the unit triangle with
boundary points,
interior points, and
area.
Solution by a1b2