Comparing Volumes Computed by Prismoidal and Trapezoidal Formula in Earthwork

Prismoidal and trapezoidal formulas are two methods of computing the volume of earthwork, which is the amount of excavated or filled material in a construction project. In general, the prismoidal formula is more accurate than the trapezoidal formula because it takes into account the curvature of the cross-sections, while the trapezoidal formula assumes that each cross-section is a flat trapezoid.

However, when comparing the volumes computed by these two formulas, the result may vary depending on the shape and complexity of the cross-sections. Here are some possible scenarios:

1. Simple and symmetrical cross-sections
If the cross-sections are simple and symmetrical, such as a rectangle, a triangle, or a circular segment, both formulas should give the same volume because the prismoidal correction factor reduces to 1. In this case, the answer is (d) same.

2. Irregular and non-symmetrical cross-sections
If the cross-sections are irregular and non-symmetrical, such as a ditch, a channel, or a road embankment, the prismoidal formula will usually give a greater volume than the trapezoidal formula. The reason is that the prismoidal formula accounts for the bulging or sagging of the middle ordinate, which is the average width of the cross-section. The trapezoidal formula assumes that the middle ordinate is the same as the average width, which is not true for most non-prismatic shapes. Therefore, the prismoidal formula is more accurate in estimating the true volume of earthwork. In this case, the answer is (b) greater.

3. Composite cross-sections
If the cross-sections are composite, meaning they consist of multiple shapes or segments, both formulas may give different volumes, depending on the distribution and arrangement of the parts. In this case, the prismoidal formula will still be more accurate than the trapezoidal formula, but the difference may not be as significant as in scenario 2. The answer could be either (a) smaller or (b) greater, depending on the specific case.

In conclusion, while computing the volume of earthwork, the prismoidal formula should be preferred over the trapezoidal formula for its greater accuracy, especially for irregular and non-symmetrical cross-sections. However, the comparison of volumes computed by these formulas should be done with caution and consideration of the shape and complexity of the cross-sections.