Difference between revisions of "Pick's Theorem"
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It is similar to the [[Shoelace Theorem]], and although it is less powerful it is a good tool to have in solving problems. | It is similar to the [[Shoelace Theorem]], and although it is less powerful it is a good tool to have in solving problems. | ||
| − | {{ | + | <asy> |
| + | size(150); | ||
| + | defaultpen(linewidth(0.8)); | ||
| + | for (int i = 1; i <= 5; i=i+1) { | ||
| + | for (int j = 1; j <= 5; j=j+1) { | ||
| + | dot((i,j)); | ||
| + | } | ||
| + | } | ||
| + | draw((1,1)--(1,3)--(3,4)--(2,5)--(5,5)--(2,2)--(4,1)--cycle);</asy> | ||
== Proof == | == Proof == | ||
Revision as of 20:08, 4 April 2012
Pick's Theorem expresses the area of a polygon, all of whose vertices are lattice points in a coordinate plane, in terms of the number of lattice points inside the polygon and the number of lattice points on the sides of the polygon. The formula is:
where 
is the number of lattice points in the interior and 
being the number of lattice points on the boundary.
It is similar to the Shoelace Theorem, and although it is less powerful it is a good tool to have in solving problems.
![[asy] size(150); defaultpen(linewidth(0.8)); for (int i = 1; i <= 5; i=i+1) { for (int j = 1; j <= 5; j=j+1) { dot((i,j)); } } draw((1,1)--(1,3)--(3,4)--(2,5)--(5,5)--(2,2)--(4,1)--cycle);[/asy]](http://latex.artofproblemsolving.com/5/5/c/55ced6a62b824d8f7b144136c14f6fe9580d9ad4.png)
Proof
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