Q2. Which of the following methods is used to calculate the area between irregular boundaries?
📚 Detailed Explanation: Simpson’s Rule for Irregular Boundary Areas
In surveying, the boundary of a plot or field is rarely a straight line. When the boundary is irregular or curved, standard geometric formulas fail. Simpson’s rule is the standard method for computing the area between an irregular boundary and a base line, using regularly spaced perpendicular offsets measured from the base line to the boundary.
Area = (d/3) × [(O&sub1; + Oₙ) + 4(O&sub2; + O&sub4;) + 2(O&sub3;)]
Where d = common interval between offsets, O&sub1;, O&sub2;, … = successive offset lengths.
Key requirement: The number of divisions must be even (i.e., the total number of offsets must be odd).
How Simpson’s Rule Works
The method assumes that the boundary curve between any three consecutive offset points forms a parabolic arc rather than a straight line. Fitting parabolas instead of straight lines captures the curvature of the boundary more accurately than the trapezoidal approximation, which assumes straight-line segments between offsets.
Why the Other Options Are Wrong
A (Geometric method): The geometric method (dividing the area into triangles, rectangles, trapezoids) works only when the boundary consists of straight lines with known dimensions. It cannot handle truly irregular or curved boundaries without multiple approximations.
B (Departure and total latitude): This method computes the area of a closed traverse polygon from the coordinates of the traverse stations. It works for polygons whose vertices are known points, not for areas with continuously irregular curved boundaries between measured offsets.
C (Double parallel distance, DPD): The DPD method is an alternative computation of traverse area using latitudes and double parallel distances. Like the latitude/departure method, it applies to traverse polygons, not to irregular boundary offset measurements.
Comparison of Area Methods
| Method | Best Used When | Boundary Type |
|---|---|---|
| Simpson’s rule | Offsets to an irregular curved boundary | Curved / irregular |
| Trapezoidal rule | Offsets to a boundary (less accurate) | Gently curved |
| Geometric / coordinate | Straight-sided polygon with known vertices | Straight-sided |
| Latitude / departure | Closed traverse with surveyed legs | Any traverse polygon |
Key Concepts for Students
- Odd number of offsets required: Simpson’s rule works in pairs of intervals (three offsets per pair). If you have an even number of offsets, the last interval must be handled separately — typically with the trapezoidal rule — and the results added.
- Parabolic vs linear assumption: Simpson’s rule uses a parabola through three points; the trapezoidal rule uses a straight line through two. The parabolic assumption is more accurate for smoothly curved natural boundaries and is why Simpson’s rule outperforms the trapezoidal rule for irregular areas.
- Exam tip: If a question mentions “irregular boundary” or “curved boundary” in the context of offset measurements, the answer is almost always Simpson’s rule. Departure/latitude methods appear in traverse area questions, not offset-based irregular boundary questions.