In civil engineering, contour maps are often used to determine the volume of earthwork involved in excavations or embankments. The calculation is based on the areas enclosed by different contour lines and the vertical interval between them. Two commonly used methods are the Trapezoidal Formula and the Prismoidal Formula.

In this example, we are tasked with calculating the volume of a hill using contour data taken at 3-meter intervals.

Given Data

• Contour interval, D = 3 m (difference between successive contours)

• Areas enclosed by contours:

  • A₀ = 375 m²

  • A₁ = 620 m²
  • A₂ = 945 m²
  • A₃ = 1365 m²
  • A₄ = 1880 m²
  • Aₙ = 2570 m² (last contour)

1. Using the Trapezoidal Formula

Formula:
Volume = D × [ (A₀ + Aₙ) ÷ 2 + (A₁ + A₂ + … + Aₙ₋₁) ]

Calculation:
Volume = 3 × [ (375 + 2570) ÷ 2 + (620 + 945 + 1365 + 1880) ]
Volume = 3 × [ 1472.5 + 4810 ]
Volume = 3 × 6282.5
Volume = 18,847.5 m³

Result (Trapezoidal Formula): 18,847.5 m³

2. Using the Prismoidal Formula

Formula:
Volume = (D ÷ 3) × [ A₀ + Aₙ + 4 × (sum of even-numbered areas) + 2 × (sum of odd-numbered areas) ]

Calculation:
Volume = (3 ÷ 3) × [ (375 + 2570) + 4 × (620 + 1365) + 2 × (945 + 1880) ]
Volume = 1 × [ 2945 + 7940 + 5650 ]
Volume = 1 × 16,535
Volume = 16,535 m³

Result (Prismoidal Formula): 16,535 m³

Conclusion

• Trapezoidal Formula Volume = 18,847.5 m³
• Prismoidal Formula Volume = 16,535 m³

The prismoidal formula generally provides more accurate results when there are many contour intervals with small spacing, as it accounts for curvature more effectively. The trapezoidal formula, on the other hand, is simpler and works better when the number of contour data points is limited and the intervals are large.

In practical applications, the choice of formula depends on the level of accuracy required and the nature of the terrain.

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