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Thursday, December 11, 2014

Throwback Fact of the Week - Pick's Theorem - 12/11/14

Pick's Theorem is a clever way to calculate the area of a simple polygon. All you need is a pencil and some graphing paper to see it come to life and simplify some potentially monotonous calculations.

The method is as follows:

1. Begin by drawing a simple polygon on an equally spaced grid (i.e. graphing paper) so that all its vertices lie on grid points.
2. Count the number of grid points, i, located in the interior of the polygon. Then, count the number of grid points, b, that fall on the boundary of the polygon.
Pick's Theorem tells us that the area (in units squared), A, of the polygon can be calculated neatly as:

A = i + b/2 - 1.

Note that this Theorem applies only to simple polygons, those with no holes and consisting only of one piece.
 i = 7, b = 8A = 7+8/2-1 = 10
A useful and handy application of this theorem is roughly estimating an area on a map (of say a region or country), by overlaying a grid and using a polygon to approximate the shape of the region you are interested in.