Signed Area of a Polygon

1

2

3

-2

-1

0

1

2

-2

-1

0

1

2

signed area = 2

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.