Recently, I was shown an amazing way to estimate the areas of shapes using something called Pick's Theorem. This theorem, created by George Alexander Pick in 1899, uses a plane with points each one unit apart, basically a Cartesian Plane in a different format, and doesn't require axes. This Theorem uses the number of points seen on the interior of a shape, and points seen on the edges of a shape. "i" represents interior, or lattice, points, and "b" represents edge, or boundary, points. The formula goes as follows: Area = i + (b/2) -1. There is also a version which incorporates interior spaces in the shape, which you can read about on this website. This theorem is especially useful for finding the areas of complicated shapes whose areas would be difficult to solve via rectification or by splitting the shape into several different shapes. Picks Theorem is also very similar to another method of estimating area, which uses a Cartesian Plane format with a grid rather than just points, in which you count the number of squares within a shape, and then count the number of squares divided or cut by the shape. Using Picks variables, "i" would be the interior squares, and "b" would be the boundary squares. By cutting the number of cut squares in half and adding that number to whole, interior squares, you can estimate the area of an object. The formula would be A = i + b/2. Below are some examples of shapes whose areas were estimated using both methods.
Picks Theorem: Area = Lattice Points + (Boundary points/2)-1
Other Method: Area = Full Squares + Partial Squares/Two
Yellow Shape (Pick's)
A = 5 + (5/2) - 1
A = 15/2 - 1
A = 13/2
A = 6.5
Other method
A = 2 + (10/2)
A = 2 + 5
A = 7
Light blue shape
A = 2 + (9/2) -1
A = 13/2 -1
A = 11/2
A = 5.5
Other method
A = 0 + (11/2)
A = 5.5
Red Square
A = 0 + (4/2) -1
A = 2 -1
A = 1
Other method
A = 1 + (0/2)
A = 1
Hexagon
A = 4 + (6/2) -1
A = 4 + 3 -1
A = 6
Other method
A = 3 + 6/2
A = 3 + 3
A =6
Gray Triangle:
A =1+(3/2)-1
A = 3/2
A = 1.5
Other method
A = 0 + 4/2
A = 2
Second green triangle:
A = 0+(5/2)-1
A = 3/2
A = 1.5
Other Method
A = 0 +3/2
A = 1.5
Green shape(R):11+(11/2)-1=15 1/2=15.5
Figure A
A = 25 + (13/2) -1
A = 25 +6.5 -1
A = 31.5
Other Method
A = 22+½(14)
A = 22 + 7
A = 31
Figure Q
A = 10 + (12/2) - 1
A = 10 + 6 -1
A = 15
Other method
A = 7+½(14)
A = 7 + 7
A = 14
Figure R
A = 9 + 12/2
A = 9 + 6
A = 15
Other Method
A = 9 + ½13
A = 9 + 6.5
A = 15.5
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