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Shoelace Formula


The shoelace formula, also known as Gauss's area formula, the shoelace algorithm, shoelace method, or surveyor's formula, is a name sometimes given to the polygon area formula for the area of a simple polygon in terms of the Cartesian coordinates of its vertices (x_1,y_1), ..., (x_n,y_n).

PolygonArea

The area of such a polygon is

 A=1/2(|x_1 x_2; y_1 y_2|+|x_2 x_3; y_2 y_3|+...+|x_n x_1; y_n y_1|),
(1)

where |M| denotes a determinant. When recast in abbreviated form (which is actually an abuse of determinant notation) as

A=1/2|x_1 x_2 ... x_n x_1; y_1 y_2 ... y_n y_1|
(2)
=1/2|x_1 y_1; x_2 y_2; | |; x_n x_n; x_1 y_1|,
(3)

the result is sometimes known as the shoelace formula due to the placement of terms with alternating signs resembling the alternating crossings of laces on the front of a show.


See also

Determinant, Polygon Area, Simple Polygon

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Cite this as:

Weisstein, Eric W. "Shoelace Formula." From MathWorld--A Wolfram Web Resource. https://mathworld.wolfram.com/ShoelaceFormula.html

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