Area of a Polygon
Area of a Polygon
The (signed) area of a planar polygon with vertices , , …, that is not self-intersecting is
(,)
x
1
y
1
(,)
x
2
y
2
(,)
x
n
y
n
1
2
x
1
y
2
x
2
y
1
x
2
y
3
x
3
y
2
x
n-1
y
n
x
n
y
n-1
x
n
y
1
x
1
y
n
1
2
x 1 |
y 1 |
x 2 |
y 2 |
x 2 |
y 2 |
x 3 |
y 3 |
x n-1 |
y n-1 |
x n |
y n |
x n |
y n |
x 1 |
y 1 |
To calculate the formula, form a three-column matrix of rows , , , ... , .
A=(,)
x
1
y
1
B=(,)
x
2
y
2
C=(,)
x
3
y
3
A=(,)
x
1
y
1
To this matrix append the column +, , , ..., , and the column , , , ..., , .
x
1
y
2
x
2
y
3
x
n-1
y
n
x
n
y
1
-
-
x
2
y
1
-
x
3
y
2
-
x
n
y
n-1
-
x
1
y
n
The area is the arithmetic mean of the sum of the terms in the "+" column and the sum of of the terms in the "" column.
-