What is Pick's theorem and how to apply?

The theorem of Pick says, that if  a polygon encloses the grid of equal distance points and if the i  is the number of interior points and b is the number of points on the boundary, then the area of the polygon is

(i+b/2-1) square units.

Application: The area of a polgon could be assesed. Construct the polygon to the scale on a graph sheet and we can calculate the area of the polygon on the graph sheet. Then the area could be squarely proportional to the area found on the graph by using Pick's theorem.

Pick's theorem allow calculation of a polygon area in Cartesian plane,whose peaks are integer coordinates. It is an elegant result, apparently, little known to us. Here's the exact statement of Pick's theorem :

In a Cartesian system is considered a polygon whose vertices have integer coordinate. Polygon area is then given by:

A=G + (1/2)*  -1, where G is the number of points inside polygon with integer coordinates, and BETA is the number of points with integer coordinates, which belongs to polygon.

Pick's theorem is true if the polygon is a triangle or a rectangle,  whose sides are parallel to coordinate axes.

Polygon is a rectangle with sides parallel to coordinate axes. If the rectangle, ABCD noted, contains points inside the side AB, with integer coordinates 'a' and points with integer coordinate  "b", within the side BC, then the length AB is equal to(a+ 1), length of side BC is (b +1). So, area of the rectangle ABCD is (a +1)* (b +1)

G + (1/2)*  -1=a*b + (1/2)*(2a+2b+4)-1=a*b+a+b-1=

= (a +1)* (b +1)=Area of ABCD Q.E.D.