A (simple) polygon polygon is a two-dimensional (2D),
closed geometric figure without lines crossing.
Pick's theorem —named
after Austrian mathematician Georg Alexander
Pick— is a theorem about simple polygons that says the following:
If a simple polygon has only whole number-valued points as its (border) points, then let
a: the number of whole number-valued points interior to the polygon
b: the number of whole number-valued border points (including all implied whole number-valued points)
then the area enclosed by the polygon equals a + (b / 2) - 1.
The following are two examples:
Example 1: unit square: a = 0; b = 4 —> area = 0 + (4 / 2) - 1 = 1
Example 2:
a (red) = 1; b (green) = 96 —> Area = 1 + (96 / 2) - 1 = 48
We can play with this:
In the grid below, construct your own (simple) polygon by picking polygon points on
the intersections of vertical and horizontal lines. To close (complete) a polygon, select the starting point.
Please note that a (simple) polygon is a closed figure (starting and end points are the same),
of which the line segments do not cross and do not 'double back' on themselves; i.e., angles of 180%
between points are not allowed.
