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 $(a_1, b_1)$, $(a_2, b_2)$, ... , $(a_n, b_n)$. The area of the polygon is

\[\dfrac{1}{2} |a_1b_2+a_2b_3+\cdots +a_nb_1-b_1a_2-b_2a_3-\cdots -b_na_1|.\]

Shoelace Theorem gets it's name by listing the coordinates like so:

\[(a_1, b_1)\] \[(a_2, b_2)\] \[\vdots\] \[(a_n, b_n)\] \[(a_1, b_1)\]

Proof

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