Pick’s Theorem

Given a simple polygon constructed on a grid of equal-distanced points (i.e., points with integer coordinates) such that all the polygon’s vertices are grid points, Pick’s theorem provides a simple formula for calculating the area ‘A’ of this polygon in terms of the number i of lattice points in the interior located in the polygon and the number b of lattice points on the boundary placed on the polygon’s perimeter:

A=i+{\frac {b}{2}}-1.

Pick-theorem-example

In the example shown, we have i = 7 interior points and b = 8 boundary points, so the area is A = 7 + 8/2 − 1 = 7 + 4 − 1 = 10 square units.

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