determine the area of sections of this kind, you shoulduse a method of determining area by coordinates.For explanation purpose, let’s consider station 305(fig. 10-6). First, consider the point where the centerline intersects the grade line as the point of origin forthe coordinates. Vertical distances above the grade lineare positive Y coordinates; vertical distances below thegrade line are negative Y coordinates. A point on thegrade line itself has a Y coordinate of 0. Similarly,horizontal distances to the right of the center line arepositive X coordinates; distances to the left of the centerline are negative X coordinates; and any point on thecenter line itself has an X coordinate of 0.Plot the cross section, as shown in figure 10-7, andbe sure that the X and Y coordinates have their propersigns. Then, starting at a particular point and goingsuccessively in a clockwise direction, write down thecoordinates, as shown in figure 10-8.After writing down the coordinates, you then mul-tiply each upper term by the algebraic difference ofthe following lower term and the preceding lower term,as indicated by the direction of the arrows (fig. 10-8).The algebraic sum of the resulting products is thedouble area of the cross section. Proceed with thecomputation as follows:Since the result (1,080.70 square feet) representsthe double area, the area of the cross section is onehalf of that amount, or 540.35 square feet.By similar method, the area of the cross section atstation 306 (fig. 10-7) is 408.40 square feet.EARTHWORK VOLUME.— As discussedpreviously, when you know the area of two crosssections, you can multiply the average of thosecross-sectional areas by the known distance betweenthem to obtain the volume of earth to be cut or filled.Consider figure 10-9 that shows the plotted crosssections of two sidehill sections. For this figure, whenyou multiply the average-end area (in fill) and theaverage-end area (in cut) by the distance between thetwo stations (100 feet), you obtain the estimatedamount of cut and fill between the stations. In thiscase, the amount of space that requires filling iscomputed to be approximately 497.00 cubic yards andthe amount of cut is about 77.40 cubic yards.MASS DIAGRAMS.— A concern of the highwaydesigner is economy on earthwork. He wants to knowexactly where, how far, and how much earth to movein a section of road. The ideal situation is to balancethe cut and fill and limit the haul distance. A techniquefor balancing cut and fill and determining theFigure 10-9.—Plots of two sidehill sections.Figure 10-8.—Coordinates for cross-section station 305 shown in figure 10-7.10-10