Shoelace Theorem
The Shoelace Theorem is a nifty formula for finding the area of a polygon given the coordinates of it's vertices.
Theorem
Suppose the polygon 
has vertices 
, 
, ... , 
, listed in clockwise order. Then area of 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)
The Shoelace Theorem gets its name because if one lists the the coordinates in a column,
and marks the pairs of coordinates to be multiplied, the resulting image looks like laced-up shoes.
Proof
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