 TOPICS # Polygon Area

The (signed) area of a planar non-self-intersecting polygon with vertices , ..., is (1)

where denotes a determinant. This formula is sometimes written in an abbreviated form as   (2)   (3)

which, while an abuse of determinant notation, is known as the shoelace formula. This can be written   (4)   (5)

where the endpoints are defined as and . 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).

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

## Explore with Wolfram|Alpha More things to try:

## 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.

Polygon Area

## Cite this as:

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