The shoelace formula, also known as Gauss's area formula, the shoelace algorithm, shoelace method, or surveyor's formula, is a name sometimes given to the polygon
area formula for the area of a simple polygon
in terms of the Cartesian coordinates of its vertices 
, ..., 
.
The area of such a polygon is
 |
(1)
|
where 
denotes a determinant. When recast in abbreviated
form (which is actually an abuse of determinant notation)
as
 |  |  |
(2)
|
 |  |  |
(3)
|
the result is sometimes known as the shoelace formula due to the placement of terms with alternating signs resembling the alternating crossings of laces on the front of a show.
