Area by planimeter

Custom Search

Area by trapezoidal formula

Intermediate Engineering

Plotting Horizontal Control

AREA BY PLANIMETER. A planimeter is a mechanical device that you can use to compute the area of an irregular figure after tracing the perimeter of a scale drawing of the figure with the tracing point on the planimeter. The most commonly used instrument is called the polar planimeter.

Figure 7-30 shows a polar planimeter. Its parts include an anchor point, P; a tracing point, T, with a guide, G; a vernier, V; and a roller, R. An adjustable arm, A, is graduated to permit adjustment to conform to the scale of the drawing. This adjustment provides a direct ratio between the area traced by the tracing point and the revolutions of the roller. As the tracing point is moved over the paper, the drum, D, and the disk F, revolve. The disk records the revolutions of the roller in units and

Figure 7-30.Polar planimeter.

Figure 7-31.Area within straight-line and curved-line boundaries (curved segments).

tenths; the drum, in hundredths; and the vernier, in thousandths.

Specific instructions for using the polar planimeter are found in the instruction booklet that is provided with the instrument. With minimal practice, you will find that the planimeter is a simple instrument to operate. You should remember, though, that the accuracy obtained with the planimeter depends mostly on the skill of the operator in accurately tracing the boundary lines of the figure with the tracing point of the planimeter.

If the instruction booklet has been lost, do not worry. The planimeter can still be used. Simply determine how many revolutions of the roller it takes to trace a figure of known area (drawn to the same scale as the figure you wish to determine the area of). Then trace the figure you are working with and read the number of revolutions taken to trace the unknown area. You now know three values as follows: (1) the area of the figure of known size, (2) the number of revolutions taken to trace the figure of known size, and (3) the number of revolutions taken to trace the figure of unknown size. B y ratio and proportion, you can then determine the unknown area.

PARCELS THAT INCLUDE CURVES. Not all parcels of land are bounded entirely by straight lines. You may have to compute the area of a construction site that is bounded in part by the center lines or edges of curved roads or the right-of-way lines of curved roads. Figure 7-31 shows a construction site with a shape similar to the traverse you have been studying in previous examples. In this site, however, the traverse lines AB and CD are the chords of circular curves, and the boundary lines AB and CD are the arcs intercepted by the chords. The following sections explain the method of determining the area lying within the straight-line and curved-line boundaries.

The data for each of the curves is inscribed on figure 7-31; that is, the radius R, the central angle A, the arc length A (to be discussed in chapter 11 of this

Figure 7-32.Computation of area which includes curve segments.

TRAMAN), the tangent length T and the chord bearing and distance C_h.

The crosshatched areas lying between the chord and arc are called segmental areas. To determine the area of this parcel, you must (1) determine the area lying within the straight-line and chord (also straight-line) boundaries, (2) determine the segmental areas, (3) subtract the segmental area for Curve 1 from the straight-line boundary area and (4) add the segmental area for Curve 2 to the straight-line boundary area. The method of determining a segmental area was explained in the EA3 TRAMAN. The straight-line area may be determined by the coordinate method, as explained in this chapter. For figure 7-31, the segmental area for Curve 1 works out to be 5,151 square feet; for

Curve 2, it is 29,276 square feet.

Figure 7-32 shows atypical computation sheet for the area problem shown in figure 7-31. Included with the station letter designations in the station column are designations (Chord #1 and Chord #2) showing the bearings and distances that constitute the chords of Curves 1 and 2. The remainder of the upper part of the form shows the process (with which you are now familiar) of determining latitudes and departures from the bearings and distances, coordinates from the latitudes and departures, double areas from cross multiplication of coordinates, double areas from the difference between the sums of north and sums of east coordinates, and areas from half of the double areas. As you can see in figure 7-32, the area within the straight-line boundaries is 324,757 square feet. From this area, segmental area No. 1 is subtracted. Then segmental area No. 2 is added.

To obtain the area of the parcel as bounded by the arcs of the curves, you must add or subtract the segmental areas depending on whether the particular area in question lies inside or outside of the actual curved boundary. In figure 7-31, you can see that the segmental area for Curve 1 lies outside and must be subtracted from the straight-line area, while that for Curve 2 lies inside and must be added. With the segmental areas accounted for, the area comes to 348,882 square feet or 8.01 acres.

The second method of determining a curved-boundary area makes use of the external areas rather than the segmental areas of the curves, as shown in figure 7-33. The straight-line figure is defined by the tangents of the curves, rather than by the chords. This method may be used as an alternative to the chord method or to check the result obtained by the chord method.

The computation sheet shown in figure 7-34 follows the same pattern as the one shown in figure 7-32.

However, there are two more straight-line boundaries,

Figure 7-33.Area within the curve and its tangents.

Figure 7-34.Computation of area which includes external area of curves.

in this case, because each curve has two tangents rather than a single long chord.

The coordinates of A, B, C, and D are the same as in the first example, but the coordinates of the points of intersection (PIs) must be established from the latitudes and departures of the tangents. The computations for determining the tangent bearings are shown in the lower left of figure 7-34. When you have only the chord bearing, you can compute the tangent bearing by adding or subtracting one half of delta (A) as correct. The angle between the tangent and the chord equals N2.

After setting coordinates on the PIs, you cross-multiply, accumulate the products, subtract the smaller from the larger, and divide by 2, as before, to get the area of the straight-line figure running around the tangents. You then add or subtract each external area as appropriate. In figure 7-33, you can see that the external area for Curve 1 is inside the parcel boundary and must be added, while that of Curve 2 is outside and must be subtracted. The area comes to 348,881 square feet, which is an acceptable check on the area obtained by using segmental areas.