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  "body": "\n## Statement\n\nLet $P$ be a simple [[polygon|8VEFWD3EJB]] in the plane whose vertices all have integer coordinates.\nLet $i$ be the number of points of $\\Z^2$ in the interior of $P$ and $b$ the number of points of $\\Z^2$ on its boundary.\nThen the area $A$ of the region bounded by $P$ is\n$$A = i + \\frac{b}{2} - 1.$$\n\n## Remarks\n\nBoth sides are additive when two lattice polygons are glued along a common edge, once the shared boundary points are counted correctly.\nSo it suffices to check the formula for lattice triangles, and then for triangles with no lattice points other than their vertices, which have area $1/2$.\nExample: the triangle with vertices $(0, 0), (2, 0), (0, 2)$ has area $2$, $b = 6$ and $i = 0$, and $0 + 3 - 1 = 2$.\nSimplicity matters: for a region with holes the formula changes by a term for each hole.\nThere is no direct analogue in dimension $3$: the tetrahedra with vertices $(0, 0, 0), (1, 0, 0), (0, 1, 0), (1, 1, r)$ have no other lattice points but volume $r/6$.\nThe theorem is due to Georg Pick.\n\n## Explanations\n\n### L1 human\n\nDraw a polygon on squared paper with its corners on grid points.\nTo find its area, count the grid points strictly inside and the grid points on the edges.\nThe area is the inside count plus half the edge count, minus one.\n\n### L2 human\n\nPick's theorem links a continuous quantity, area, to a count of lattice points.\nIt is the two-dimensional seed of Ehrhart theory, which counts lattice points in dilates of lattice polytopes.\nIt also gives a quick proof that a lattice triangle with no other lattice points has area $1/2$, which is used in the study of Farey sequences.\n",
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