Difference between revisions of "Pick's Theorem"

(it's a theorem. ;))
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'''Pick's Theorem''' expresses the [[area]] of a [[polygon]], all of whose [[vertex | 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:
 
'''Pick's Theorem''' expresses the [[area]] of a [[polygon]], all of whose [[vertex | 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:
  

Revision as of 00:30, 14 September 2008

This is an AoPSWiki Word of the Week for September 11- September 18

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:

$A = I + \frac{B}{2} - 1$

where $I$ is the number of lattice points in the interior and $B$ being the number of lattice points on the boundary. It is similar to the shoestring formula, and though it is less powerful it is a good tool to have in solving problems.


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Proof

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