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

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

PolygonArea

This can be written

A=1/2sum_(i=1)^(n)(x_iy_(i+1)-x_(i+1)y_i)
(4)
=1/2(x_1y_2-x_2y_1+x_2y_3-x_3y_2+...+x_(n-1)y_n-x_ny_(n-1)+x_ny_1-x_1y_n),
(5)

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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References

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. http://paulbourke.net/geometry/polygonmesh/.Nürnberg, R. "Calculating the Area and Centroid of a Polygon in 2D." 2013. https://www.ma.imperial.ac.uk/~rn/centroid.pdf.

Referenced on Wolfram|Alpha

Polygon Area

Cite this as:

Weisstein, Eric W. "Polygon Area." From MathWorld--A Wolfram Web Resource. https://mathworld.wolfram.com/PolygonArea.html

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