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.