Pick's Theorem

Pick's theorem (or rule) easily calculates the area (surface) of a polygon with $ b $ vertices built on a lattice, a 2D grid of points with integer coordinates (points with equal distances). If all $ b $ vertices of the polygon (vertices can be flat) are grid points and the polygon has $ i $ points inside itself then Pick's formula indicates that the polygon area $ A $ is equal to $$ A = i + \frac{b}{2} - 1 $$

All the points present on the contour are considered as vertices.

The Pick formula requires only two parameters: the number $ i $ of interior points of the polygon and the number $ b $ of vertices of the polygon (which is in the number of grid points on the perimeter of the polygon). Thearea $ A $ of the polygon is$ A = i + \ frac {b} {2} - 1 $

Example: The polygon drawn below has 15 points inside the polygon (light gray), and 10 vertices (dark gray). Its area is therefore $ A = 15 + 10/2 - 1 = 19 $.

The formula owes its name to Georg Alexander Pick who described it in 1899.