METBD 050
Coordinate Method:
The area of any polygon (regular or irregular – see Figure 1) can be determined using a method called the
“Coordinate method.” In this method, the coordinates of the vertices are systematically multiplied, then
added or subtracted to obtain the area bounded by the polygon.
Figure 1: Polygons
METHOD:
1. List the x and y-coordinates of each of the vertices in a column, repeating the starting point
coordinate at the end of the list. The coordinates have to be listed in the order they occur while
tracing around the polygon in either the clockwise or counterclockwise direction.
2. Multiply each x value by the next y-value in the list and add the products.
 x (y
i
i 1 )
3. Multiply each y-value by the next x-value in the list and add the products.
 y (x
i
i 1 )
4. Obtain the area by evaluating the expression:
Area 
1
2
 x (y
i
i1 ) 
 y (x
i
i1 )
EXAMPLE: Find the area bounded by a polygon having 5 sides and coordinates (X 1, Y1) through (X5, Y5).
List of Coordinates
First Products
Second Products
X
Y
Xi(Yi+1)
Yi(Xi+1)
X1
Y1
X1Y2
Y1X2
X2
Y2
X2Y3
Y2X3
X3
Y3
X3Y4
Y3X4
X4
Y4
X4Y5
Y4X5
X5
Y5
X5Y6
Y5X6
X6 = X1
Y6 = Y1
X1Y2+ X2Y3+ X3Y4+ X4Y5+ X5Y6
Y1X2+ Y2X3+ Y3X4+Y4X5+ Y5X6
Sum =
Area = (1/2) | X1Y2+ X2Y3+ X3Y4+ X4Y5+ X5Y6 – (Y1X2+ Y2X3+ Y3X4+Y4X5+ Y5X6) |
This method truly lends itself to a solution using a spreadsheet.
METBD 050 – Coordinate Method.doc
1/30/02, Revised 10/1/03
Page 1 of 2
EXAMPLE: Find the area bounded by the polygon shown in the figure below using method of Area by
Coordinates.
Y
(3,7)
(5, 7)
(3,5)
(1,4)
(6,3)
(1,1)
(4,1)
X
List of Coordinates
X
Y
1
1
3
3
5
6
4
1
1
4
5
7
7
3
1
1
Sum =
First Products
Second Products
Xi(Yi+1)
Yi(Xi+1)
1(4) = 4
1(5) = 5
3(7) = 21
3(7) = 21
5(3) = 15
6(1) = 6
4(1) = 4
1(1) = 1
4(3) = 12
5(3) = 15
7(5) = 35
7(6) = 42
3(4) = 12
1(1) = 1
4+5+21+21+15+6+4 = 76
1+12+15+35+42+12+1 = 118
|
|
|
|
Area = (1/2) 76 – 118 = (1/2) -42 = 21
CHECK: Divide area into shapes having well-known area formulas:
Y
(3,7)
(3,5)
(5, 7)
6
2
(1,4)
1
1
A = 9+1+6+2+1+2 = 21
(6,3)
9
Checks
2
(1,1)
The area of the well-known
shapes are shown in circles.
(4,1)
X
METBD 050 – Coordinate Method.doc
1/30/02, Revised 10/1/03
Page 2 of 2