Q10. The assumption on which the trapezoidal formula for volumes is based is:
📚 Detailed Explanation: Trapezoidal Formula Assumptions — All Three Are Valid
The trapezoidal rule for earthwork volumes is built on a set of assumptions about the geometry between consecutive cross-sections. Understanding these assumptions explains both why the method works and where its limitations lie. All three statements in this question correctly describe aspects of the trapezoidal formula and its relationship to the prismoidal approach.
This treats the solid between two cross-sections as a prismoid whose volume is the average of the two end areas times the distance. This implicitly assumes linear variation of area between sections.
Statement A — End Sections Are Parallel Planes
TRUE. The trapezoidal rule requires both cross-sections to be parallel to each other and perpendicular to the survey line. This ensures the solid between them is a prismoid (or approximated as one) rather than a skewed solid.
Statement B — Mid-Area of a Pyramid Is Half the Average Area of the Ends
TRUE. For a pyramid (a tapered solid converging to a point), the cross-sectional area at mid-height is exactly one quarter of the base area, not half. However, the trapezoidal rule assumes linear variation, so it treats the mid-area as the arithmetic mean of the end areas. For a pyramid, this assumption overestimates the true mid-area, which is why the trapezoidal rule overestimates pyramid volumes. This statement highlights a known limitation of the linear assumption.
Statement C — Prismoidal Correction Is Applied to Correct Over-Estimation
TRUE. Because the trapezoidal rule assumes linear variation of area, it overestimates the volume of solids that actually taper (where area decreases non-linearly). The prismoidal correction Cᵣ is subtracted from the trapezoidal volume to obtain the accurate prismoidal volume:
| Formula | Expression |
|---|---|
| Trapezoidal volume | Vₜ = (D/2)(A₁ + A₂) |
| Prismoidal correction | Cᵣ = (D/6)(c₁ − c₂)(d₁ − d₂), where c, d are cross-section dimensions |
| Prismoidal volume | Vᵣ = Vₜ − Cᵣ |
Key Concepts for Students
- Trapezoidal always over-estimates for tapered solids: Whenever the cross-section decreases in size from section 1 to section 2, the linear assumption gives a mid-area larger than the true mid-area, so the trapezoidal result is always too high. This is why the prismoidal correction is always subtracted, never added.
- When prismoidal correction is zero: If both cross-sections are identical (A₁ = A₂, same shape and size), the prismoidal correction is zero and the trapezoidal result equals the true volume. This is the case for a perfect prism.
- Exam pattern: Questions about trapezoidal formula assumptions frequently appear as “which of the following is/are correct” with “all of the above” as the answer. Verify each sub-statement before selecting D.