WOLFRAM|DEMONSTRATIONS PROJECT

Signed Area of a Polygon

​
The signed area of a polygon with vertices
{
x
i
,
y
i
}
numbered
1
through
n
can be calculated exactly by the formula
1
2
n
∑
i=1
(
x
i
y
i+1
-
x
i+1
y
i
)
, where
x
n+1
=
x
1
and
y
n+1
=
y
1
. If the polygon is simple (non-intersecting sides), with the vertices numbered in a counterclockwise direction, the signed area is the area. This formula is surprisingly useful in surveying, architecture, and many other applications.