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Pick's theorem (Theorem)

Let $ P\subset\mathbb{R}^2$ be a polygon with all vertices on lattice points on the grid $ \mathbb{Z}^2$. Let $ I$ be the number of lattice points that lie inside $ P$, and let $ O$ be the number of lattice points that lie on the boundary of $ P$. Then the area of $ P$ is

$\displaystyle A(P) = I + \frac{1}{2}O - 1. $
\includegraphics{pick}
In the above example, we have $ I=5$ and $ O=13$, so the area is $ A=10\frac{1}{2}$; inspection shows this is true.



"Pick's theorem" is owned by ariels.
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proof of Pick's theorem (Proof) by giri
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Cross-references: area, boundary, lie on, number, grid, points, lattice, vertices, polygon
There is 1 reference to this entry.

This is version 1 of Pick's theorem, born on 2002-06-12.
Object id is 3096, canonical name is PicksTheorem.
Accessed 10748 times total.

Classification:
AMS MSC: 51A99 (Geometry :: Linear incidence geometry :: Miscellaneous)
 05B99 (Combinatorics :: Designs and configurations :: Miscellaneous)
 68U05 (Computer science :: Computing methodologies and applications :: Computer graphics; computational geometry)
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no holes by akrowne on 2002-06-12 16:47:52
Also note that our definition of "polygon" here excludes holes (i.e. no annular regions, only regions of genus 0.) The theorem will actually fail in this case, as is easy to see in a simple 3x3 subsection of the grid with the middle square excluded.

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