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).
