Shoelace Theorem

Revision as of 08:21, 19 October 2008 by 7h3.D3m0n.117 (talk | contribs) (Theorem)

The Shoelace Theorem is a nifty formula for finding the area of a polygon given the coordinates of its vertices.

Theorem

Suppose the polygon $P$ has vertices $(a_1, b_1)$, $(a_2, b_2)$, ... , $(a_n, b_n)$, listed in clockwise order. Then area of $P$ is

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

The Shoelace Theorem gets its name because if one lists the the coordinates in a column, \begin{align*} (a_1 &, b_1) \\ (a_2 &, b_2) \\ & \vdots \\ (a_n &, b_n) \\ (a_1 &, b_1) \end{align*}, and marks the pairs of coordinates to be multiplied, the resulting image looks like laced-up shoes.

Proof

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