Polygon Area

The (signed) area of a planar non-self-intersecting polygon with vertices (x_1,y_1), ..., (x_n,y_n) 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|),

where |M| denotes a determinant. This formula is sometimes written in an abbreviated form as

A=1/2|x_1 x_2 ... x_n x_1; y_1 y_2 ... y_n y_1|
=1/2|x_1 y_1; x_2 y_2; | |; x_n x_n; x_1 y_1|

which, while an abuse of determinant notation, is known as the shoelace formula.


This can be written


where the endpoints are defined as x_(n+1)=x_1 and y_(n+1)=y_1. The alternating signs of terms can be found from the diagram above, which illustrates the origin of the term "shoelace formula."

Note that the area of a convex polygon is defined to be positive if the points are arranged in a counterclockwise order and negative if they are in clockwise order (Beyer 1987).

See also

Area, Convex Polygon, Polygon, Polygon Centroid, Shoelace Formula, Triangle Area

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Beyer, W. H. (Ed.). CRC Standard Mathematical Tables, 28th ed. Boca Raton, FL: CRC Press, pp. 123-124, 1987.Bourke, P. "Calculating the Area and Centroid of a Polygon." July 1988.ürnberg, R. "Calculating the Area and Centroid of a Polygon in 2D." 2013.

Referenced on Wolfram|Alpha

Polygon Area

Cite this as:

Weisstein, Eric W. "Polygon Area." From MathWorld--A Wolfram Web Resource.

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