Pick’s Theorem
Pick’s Theorem was first published in 1899 by Georg Alexander Pick. It gives a formula for
finding the area of a polygon placed on a grid.
The area of the polygon placed on the
centimeter grid could be found by
partitioning it into rectangles and
triangles, and adding the areas of
those.
1.5
2
6 + 12/6 + 3/2 + 3/2 +4/2 = 17
cm2.
6
6
Area = 17
1.5
The area of the same polygon
could be found using Pick’s Theorem.
Count the grid points on the boundary
of the polygon (6 red points) and the points in the interior (15 green points). Pick’s Theorem
states that the area of the polygon is one less than the sum of the interior points and half
the boundary points. So for our
example polygon,
Area
= 15 + 6/2 -1
= 15 + 3 – 1
= 17 cm2
Use Pick’s Theorem to find the areas
of the polygons below.
Office for Mathematics, Science and Technology Education
www.mste.uiuc.edu
Use Pick’s Theorem to find the area of △AGH.
Next, use Pick’s Theorem to find the areas of △’s ABH, BCH, CDH,DEH, EFH, and FGH.
(Assume a cm grid.) What do you notice?
H
A
B
C
D
E
F
Use Pick’s Theorem to find the area of each of Tangram piece.
Office for Mathematics, Science and Technology Education
www.mste.uiuc.edu
G
Draw a polygon on the grid below. Be sure each vertex is on a grid point.
Then use Pick’s Theorem to find its area. Show your work.
Office for Mathematics, Science and Technology Education
www.mste.uiuc.edu