Area from coordinates

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Latitude and departure from plane coordinates

Intermediate Engineering

Area by trapezoidal formula

AREA FROM COORDINATES. Before we explain the method of computing area from coordinates, let us set coordinates for the stations of the traverse we are working on. To avoid using negative coordinates, we will measure Y coordinates from an X axis passing through the most southerly station and X coordinates from a Y axis passing through the most westerly station, as shown in figure 7-24.

Figure 7-24.Computations of a closed traverse by coordinate method.

Figure 7-25.Coordinate entries for computation of figure 7-24.

Figure 7-26.First step for tabulated computation of figure 7-24.

Figure 7-25 shows the coordinate entries. You can see that the Y coordinate of A equals the latitude of DA, or 591.64 feet, while the X coordinate of A is zero. The Y coordinate of B equals the Y coordinate of A plus the latitude of AB or 591.64 + 255.96 = 847.60 feet.

The X coordinate of B equals the departure of AB, or 125.66 feet. The Y coordinate of C equals the Y coordinate of B minus the latitude of BC or 847.60 153.53 = 694.07 feet.

The X coordinate of C equals the X coordinate of B plus the departure of BC or 125.66 + 590.65 = 716.31 feel.

The Y coordinate of D obviously is zero; however, it computes as the Y coordinate of C minus the latitude of CD of 694.07 694.07, which serves as a check. The X coordinate of D equals the X coordinate of C minus the departure of CD or 716.31 192.69 = 523.62 feet. This is the same as the departure of DA, but with an opposite signa fact which serves as another check.

Figure 7-27.Second step for tabulated computation of figure 7-24.

Figures 7-26 and 7-27 show the method of determining the double area from the coordinates. First, multiply pairs of diagonally opposite X and Y coordinates, as shown in figure 7-26, and determine the sum of the products. Then, multiply pairs diagonally in the opposite direction, as shown in figure 7-27, and determine the sum of the products. The difference between the sums (shown in fig. 7-26) is the double area or 1,044,918.76 397,011.37 = 647,907.39 square feet The symbol shown beside the sum of the coordinate products is the capital Greek letter (S) sigma In this case, it simply means sum.