Pick's TheoremDate: 9/4/96 at 22:31:47 From: -={Mirage}=- Subject: Pick's Theorem Would you happen to know Pick's theorem? This has been bugging me for almost a year. Thanks, Jeff Date: 9/5/96 at 13:22:54 From: Doctor Tom Subject: Re: Pick's Theorem Hi Jeff, Sure. On a grid of points (like on graph paper) draw a simple polygon (no holes, no lines crossing each other), so that all the vertices lie on the grid. Then count the number of points completely inside (call this number I), and count the number of points that lie on the edges or corners of the polygon (call this number E). Then the area inside the polygon is given by the formula: I + E/2 - 1, where the area is measured in terms of the smallest squares you can make on the grid. You can find a proof of this in "Introduction to Geometry," second edition, by Coxeter (and probably in lots of other places - that's the first advanced geometry book I hauled off my shelf). Try a bunch of examples yourself to convince yourself of the truth. The proof isn't too difficult either, but it's not totally elementary, so you may have to look it up. -Doctor Tom, The Math Forum Check out our web site! http://mathforum.org/dr.math/ |
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