Pick's Theorem

 When calculating the area of a parallelogram, the normal and most well-known is to multiply base and height. But in 1899, George Alexander Pick discovered a new, innovative way of solving the area of any polygon that involves using the grid.

1.

To understand the Pick’s Theorem, the variables of the equation has to be defined. “i” is the variable for the number of interior points completely in the polygon. “b” is the variable for the number of points the polygon is touching, so basically the points on the side. Now the equation for Pick’s Theorem is A= i + (b/2) -1. So the polygon in figure 1 has 39 exterior points so i is 39. It also has 14 points that are touching the polygon so b is 14. Finally the equation is A=39+(14/2)-1

A= 39 + 7 -1

A= 45

            The following are example and the answers for Pick’s Theorem

                                              Figure 1

Green Polygon: A = 4 + (4/2)-1, A= 4+ 2-1, Area equals 5

Yellow Polygon: A =5 + (5/2)-1, A= 5 + 2.5 -1, Area equals 6.5

Blue Polygon: A = 2 + (9/2) – 1, A = 2 + 4.5 – 1, Area equals 5.5

                                                                Figure 2

Red Polygon: A = 0 + (4/2) -1, A = 0 + 2 – 1, Area is 1

Blue Polygon: A = 4 + (6/2) – 1, A = 4 + 3 – 1, Area is 6

Gray Polygon: A = 1 + (3/2) – 1, A = 1 + 1.5 -1, Area is 1.5

Green Polygon: A = 0 + (5/2) -1, A = 2.5 – 1, Area is 1.5

                                 Figure 3

For these polygons, prove that A = Q + R in more than 1 way.

A Polygon: A = 25 + (13/2) – 1, A = 25 + 6.5 – 1, Area is 30.5

Q Polygon: A = 10 + (12/2) – 1, A = 10 + 6 -1, Area is 15

R Polygon: A = 11 + (11/2) – 1, A = 11 + 5.5 -1, Area is 15.5

Q + R Polygon: Area of Q + R Polygon is 30.5, Area of A polygon is 30.5

The second way to find the area of this polygon is to make the entire polygon consist of one square. Then subtract all the shapes that are not part of the original figure.

Area of Yellow Polygon: 8 * 6= 48, 2 * 6(1/2)= 6, 2 * 2(1/2)= 2, 4 * 3 (1/2)= 6, 3 * 1 = 3, 1 * 1 (1/2)= 1/2 , 6+2+6+3+0.5=17.5. 48-17.5=Area is 30.5

Area of Purple Polygon: 6 * 7 = 42, 1 * 2(1/2)= 1, 1 * 2(1/2) = 1, 1 *1 = 1, 1 * 2(1/2)= 1, 2 * 2= 4, 1 * 1 (1/2)= ½, 1*1 (1/2= ½, 5 * 2= 10, 2 * 2(1/2)= 2, 4 * 3(1/2)= 6, 1+1+1+1+4+.5+.5+10+2+6=27, 42 -27=Area is 15.

Area of Green Polygon: 6 * 6= 36, 3 * 1= 3, 1*1(1/2)=1/2, 2*6(1/2)= 6,1 *2(1/2)=1, 1*1=1 , 1*2(1/2)=1, 3*2=6,1*2(1/2)=1,1*2(1/2)=1, 3+.5+6+1+1+1+6+1+1=20.5, 36-20.5=Area is 15.5

Area of Q + R: 15.5+15 = 30.5

Pick’s Theorem is indeed very useful because it is a formula that is able to discover the area of any polygon. This theorem can even be used for complex polygons containing holes. By using Pick’s Theorem, a simple translation for the equation of any polygon is possible.

Sources:

http://jwilson.coe.uga.edu/emat6680fa05/schultz/6690/pick/pick_main.htm