# Difference between revisions of "Shoelace Theorem"

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

− | <cmath>\dfrac{1}{2} |a_1b_2+a_2b_3+\cdots +a_nb_1-b_1a_2 | + | <cmath>\dfrac{1}{2} |(a_1b_2 + a_2b_3 + \cdots + a_nb_1) - (b_1a_2 + b_2a_3 + \cdots + b_na_1)|</cmath> |

The Shoelace Theorem gets its name because if one lists the the coordinates in a column, | The Shoelace Theorem gets its name because if one lists the the coordinates in a column, |

## Revision as of 08:21, 19 October 2008

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

## Theorem

Suppose the polygon has vertices , , ... , , listed in clockwise order. Then area of is

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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