# Pick’s 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

$$A(P)=I+\frac{1}{2}O-1.$$ |

In the above example, we have $I=5$ and $O=13$, so the area is $A=10\u2064\frac{1}{2}$; inspection shows this is true.

Title | Pick’s theorem |
---|---|

Canonical name | PicksTheorem |

Date of creation | 2013-03-22 12:46:58 |

Last modified on | 2013-03-22 12:46:58 |

Owner | ariels (338) |

Last modified by | ariels (338) |

Numerical id | 4 |

Author | ariels (338) |

Entry type | Theorem |

Classification | msc 51A99 |

Classification | msc 05B99 |

Classification | msc 68U05 |