Pick's Theorem

What is Pick's Theorem?

Pick's theorem is another method for finding the area of a polygon. This formula works as long as the polygon can be placed on a cartesian plane and the polygon's vertices has integers as coordinates. This formula was discovered in 1899 by a man named, George Alexander Pick.

Area = i + (b/2) - 1

In Pick's theorem, i represents the number of interior points  that don't connect with the polygon. b represents the boundary points. A boundary point lies on the side of the shape. So, Pick's theorem is

Area = number of interior points + (number of boundary points/2) - 1

Examples.


Green Shape:
Pick's theorem

Area = 4 + (4/2) - 1
Area = 4 + 2 -1
Area = 5 units squared

Yellow Shape:
Pick's theorem

Area = 5 + (5/2) - 1
Area = 5 + 2.5 -1
Area = 6.5 units squared

Blue Shape
Pick's Theorem

Area = 2 + (9/2) -1
Area = 2 + 4.5 -1
Area = 5.5 units squared

More Examples.




A = Q + R

Shape A

Area = 25 + (13/2) - 1
Area = 25 + 6.5 - 1
Area = 30.5 square units

Shape Q

Area = 10 + (12/2) - 1
Area = 10 + 6 - 1
Area = 15 square units

Shape R

Area = 11 + (11/2) - 1
Area = 11 + 5.5 -1
Area = 15.5 units squared

Check
A = Q + R
30.5 = 15 + 15.5
30.5 = 15.5