Difference between revisions of "Shoelace Theorem"
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Revision as of 12:05, 24 April 2008
Shoelace Theorem is a nifty formula for finding the area of a polygon given the coordinates of it's vertices.
Theorem
Let the coordinates, in "clockwise" order, be 
, 
, ... , 
. The area of the polygon is
![\[\dfrac{1}{2} |a_1b_2+a_2b_3+\cdots +a_nb_1-b_1a_2-b_2a_3-\cdots -b_na_1|.\]](http://latex.artofproblemsolving.com/0/0/f/00fb3ab55fb21e342cd434b951e5acea267f35cc.png)
Shoelace Theorem gets it's name by listing the coordinates like so:
![\[(a_1, b_1)\]](http://latex.artofproblemsolving.com/3/4/7/347b9087ed0326f4689fb1107904765d1489fc2a.png)
![\[(a_2, b_2)\]](http://latex.artofproblemsolving.com/f/1/6/f16ee4ba2c6187ee3a93abe22bdb90b036c50ff9.png)
![\[\vdots\]](http://latex.artofproblemsolving.com/7/6/4/76403adfdd35fb846675aaeaac92daec680a4fba.png)
![\[(a_n, b_n)\]](http://latex.artofproblemsolving.com/7/0/7/70770b0a072bbae13f413a562f6ea96d98503cf9.png)
Proof
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