A dot grid paper is given with irregular concave and convex polygons drown whose vertices are passing through dots.  

PS: Notice that the area of a polygon is the bordered surface by its sides. 

Definition of Pick’s theorem is: “If a polygon has vertices with integer coordinates (lattice points) then the area of the polygon is equal to: “one less than the sum of the number of interior points (dots) and half of the number of the points (dots) which lay on the borders (sides or perimeter).” 

If we write the theorem mathematically as:

AP=i+12p−1𝑨𝑷=𝒊+𝟏𝟐𝒑−𝟏; where “

AP”AP” is the area of the polygon, “i” is the number of interior dots and “p” is the number of the points which lay on the borders.