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 i1 ) y (x i i1 ) 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