Pick's Theorem to find Area of Polygon on Grid (January 24-27)

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The children did a great job last week of determining area and perimeter in terms of square units and linear units, respectively.  They did this using our City A through City D plus their own creations.

Georg Pick (1859 - 1942) was a mathematician who died in the Theresienstadt concentration camp. His simple geometry theorem allows us to find the area of any polygon that is embedded on a lattice or grid. It is especially useful for concave polygons, for which the usual textbook area formulas do not work.

Pick's Theorem works wonderfully on a geoboard or graph paper, where students can design wild polygons with rubber bands stretched over the nailheads.

Pick's Theorem Method (P/2 + I -1 = Area of Polygon)

 

First, count all of the dots on the perimeter (along the outside) of the polygon and cut that number in half (P/2 means divide by 2).

 

Second, count all of the dots in the interior (inside) of the polygon and add that number to half the number of dots in the perimeter.

 

Finally, subtract the number one (“1”) from your result above and that is the area of the polygon in square units.

 

The attached pdf file contains several crazy polygons that the children can use to practice Pick’s Theorem.  After they finish and verify their answers with my key, they should make their own and calculate the area using Pick’s Theorem.

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Picks_Theorem_Area.pdf405.87 KB