Short Answer:

The trapezoidal rule in area calculation is a method used in surveying to find the area of land with irregular boundaries. It works by dividing the total area into several trapezoids between measured offsets, then applying a specific formula to calculate the total area. This rule is simple and commonly used when the land boundary is not straight but the measurements are taken at equal intervals.

This rule assumes the boundary between two measured points to be a straight line and calculates the area accordingly. It is useful for calculating the area of curved or uneven plots when high accuracy is not required, and it gives fairly accurate results with less effort.

Trapezoidal rule in area calculation

The trapezoidal rule is one of the most widely used methods in civil surveying when dealing with irregular boundaries or curved surfaces. When the boundary line between two points is curved but is assumed to be a straight line for calculation purposes, the area is treated as a series of trapezoids. The rule helps estimate the total area between a baseline and an irregular boundary.

Method of Applying the Trapezoidal Rule

To apply the trapezoidal rule, surveyors first measure a set of offsets (distances from the baseline to the boundary) at equal intervals along a straight baseline. These offsets are then used to divide the area into several trapezoidal strips.

Let the distance between each offset be “d” and the offsets be O₁, O₂, O₃, …, Oₙ.

The formula for calculating area using the trapezoidal rule is:

Area = (d/2) × [O₁ + 2(O₂ + O₃ + … + Oₙ₋₁) + Oₙ]

This formula simply means:

• Add the first and last offsets once.
• Add all the other intermediate offsets and multiply their sum by 2.
• Add both results and multiply the final sum by half the interval length “d”.

Example Use in Surveying

Suppose there are 5 offsets taken at 10 meters apart with values: 2.0, 2.5, 3.0, 2.8, and 2.2 meters.

Using the formula:

Area = (10/2) × [2.0 + 2(2.5 + 3.0 + 2.8) + 2.2]
= 5 × [2.0 + 2(8.3) + 2.2]
= 5 × [2.0 + 16.6 + 2.2]
= 5 × 20.8 = 104 m²

This shows how the area can be quickly calculated even for irregular plots using regular interval data.

Advantages

• Simple and quick method
• Suitable for field conditions
• Requires fewer calculations than some other methods

Limitations

• Assumes straight-line boundary between offsets, which may lead to less accuracy for sharply curved lines
• Not suitable for highly irregular areas or where higher accuracy is needed (Simpson’s Rule is better in such cases)

Conclusion

The trapezoidal rule in area calculation is a simple and practical method to estimate land areas with slightly irregular boundaries. It is useful for fieldwork where speed and ease are important, although it may not be as accurate for highly curved shapes. It remains a commonly used method due to its simplicity and effectiveness.