Difference between revisions of "Shoelace Theorem"

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<cmath>(a_1, b_1)</cmath>
 
<cmath>(a_1, b_1)</cmath>
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<cmath>(a_2, b_2)</cmath>
 
<cmath>(a_2, b_2)</cmath>
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<cmath>\vdots</cmath>
 
<cmath>\vdots</cmath>
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<cmath>(a_n, b_n)</cmath>
 
<cmath>(a_n, b_n)</cmath>
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<cmath>(a_1, b_1)</cmath>
 
<cmath>(a_1, b_1)</cmath>
  

Revision as of 12:07, 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

Template:Incomplete

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