[Math Lair] Pick's Theorem

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Examples of lattice polygons

Pick's theorem is a theorem giving the area of what are called lattice polygons, polygons all of whose vertices lie on points of a point lattice. As an example, the shapes depicted at right are all lattice polygons.

Pick's theorem gives the formula for a lattice polygon's area, A, as:

A = I + ½B − 1

Looking at the examples at right:

Pick's theorem can often come in handy when you run into such polygons, although it isn't universally useful; to take one example, an equilateral triangle could never be a lattice polygon.