The (signed) area of a planar non-self-intersecting polygon with vertices , ..., is
(1)
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where denotes a determinant. This formula is sometimes written in an abbreviated form as
(2)
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(3)
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which, while an abuse of determinant notation, is known as the shoelace formula.
This can be written
(4)
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(5)
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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).