Figure 7-27.—Second step for tabulated computation of figure 7-24.Figures 7-26 and 7-27 show the method ofdetermining the double area from the coordinates. First,multiply pairs of diagonally opposite X and Ycoordinates, as shown in figure 7-26, and determine thesum of the products. Then, multiply pairs diagonally inthe opposite direction, as shown in figure 7-27, anddetermine the sum of the products. The differencebetween the sums (shown in fig. 7-26) is the double areaor 1,044,918.76 – 397,011.37 = 647,907.39 square feetThe symbol shown beside the sum of the coordinateproducts is the capital Greek letter (Z) sigma In thiscase, it simply means sum.AREA BY TRAPEZOIDAL FORMULA.— It isoften necessary to compute the area of an irregularfigure, one or more of whose sides do not forma straightFigure 7-28.—Area of irregular figure by trapezoidal rule.line. For illustration purpose, let us assume that figure7-28 is a parcel of land in which the south, east, and westboundaries are straight lines per pendicular to each other,but the north boundary is a meandering shoreline.To determine the area of this figure, first lay offconveniently equal intervals (in this case, 50.0-footintervals) from the west boundary and erect perpen-diculars as shown. Measure the perpendiculars. Call theequal interval d and the perpendiculars (beginning withthe west boundary and ending with the east boundary)hl through k.Now, you can see that for any segment lyingbetween two perpendiculars, the approximate area, bythe rule for determining the area of a trapezoid, equalsthe product of d times the average between theperpendiculars. For the most westerly segment, forexample, the area isThe total area equals the sum of the areas of thesegments; therefore, since d is a factor common to eachsegment, the formula for the total area may be expressedas follows:7-19

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