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).
