# Shoelace Theorem

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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:

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