Pick's theorem
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Georg Alexander Pick (1859–1942) discovered a remarkable way to calculate the area of certain polygons: those whose vertices lie on a regular square lattice (the points with integer coordinates in an orthonormal coordinate system). To measure their area, Pick counts the number i of lattice points inside the polygon and the number b of lattice points on its boundary—10 and 6, respectively, in the figure. The area A is equal to i + b / 2 - 1, which gives 12 here.
Check it for yourself!
To prove the formula in general, we can begin with triangles whose vertices are lattice points, then show that any polygon can be obtained by starting with such a triangle and gradually adding further triangles.
