---
title: "Area between boundary, offsets and base line"
subject: "Surveying I"
chapter: 9
key: "semester_3/surveying_i:area-offsets-chainage-0-12-20-25-34-42-52"
source: https://www.bctengineer.com/q/semester_3/surveying_i/area-offsets-chainage-0-12-20-25-34-42-52
site: BCT Engineering
---

# Area between boundary, offsets and base line

Surveying I · Chapter 9: Computation of Area and Volume

- Asked in: 2075 Asoj, 4 marks
- Video: [Offset Method: Step-by-step Area Calculation Approach](https://www.youtube.com/watch?v=R3yYgE1BvwE) — Study Force
- Answer checked by Claude Opus 5.5 (high effort) on 2026-10-14
- Page: https://www.bctengineer.com/q/semester_3/surveying_i/area-offsets-chainage-0-12-20-25-34-42-52
- Whole chapter: https://www.bctengineer.com/md/semester_3/surveying_i/chapter-9

## Question

From the chainages and offsets given below, find the area between the boundary, the first and last offsets and base line.

| Chainage (m) | 0 | 12 | 20 | 25 | 34 | 42 | 52 |
|---|---|---|---|---|---|---|---|
| Offset (m) | 0 | 6.9 | 7.6 | 9.8 | 10.2 | 9.9 | 6.8 |

## Short answer

Apply the trapezoidal rule to each strip of unequal width. Area = sum of (width × mean of adjacent offsets) = 396.80 m².

## Answer

The chainages are at unequal intervals, so the area is found strip by strip with the trapezoidal rule (the strip between two offsets is a trapezium of width equal to the chainage difference).

| Strip | Width $w$ (m) | $o_1$ (m) | $o_2$ (m) | Area $= w(o_1+o_2)/2$ (m²) |
|---|---|---|---|---|
| 1 | 12.00 | 0.00 | 6.90 | 41.4000 |
| 2 | 8.00 | 6.90 | 7.60 | 58.0000 |
| 3 | 5.00 | 7.60 | 9.80 | 43.5000 |
| 4 | 9.00 | 9.80 | 10.20 | 90.0000 |
| 5 | 8.00 | 10.20 | 9.90 | 80.4000 |
| 6 | 10.00 | 9.90 | 6.80 | 83.5000 |
| **Total** | | | | **396.8000** |

**Check by coordinates:** with the points (0,0), (52,0), (52,6.8), (42,9.9), (34,10.2), (25,9.8), (20,7.6), (12,6.9), (0,0) the formula

$$
\text{Area} = \frac{1}{2}\left|\sum \left(x_i\,y_{i+1} - x_{i+1}\,y_i\right)\right|
$$

gives 396.800 m², the same value.

**Answer:** area = **396.80 m²**.

## Check yourself

The answer's key points as multiple-choice questions, with the correct option and why.

1. Why is the area found strip by strip?
   A. The offsets are equal
   B. The boundary is straight
   C. Simpson's rule is more accurate here
   D. The chainages are at unequal intervals
   Answer: D. Unequal spacing prevents the use of Simpson's rule. (Key point: Method choice)

2. What is the area of the first strip (width 12 m, offsets 0 and 6.9)?
   A. 41.40 m2
   B. 82.80 m2
   C. 20.70 m2
   D. 55.20 m2
   Answer: A. 12 x (0 + 6.9)/2 = 41.4 m2. (Key point: Strip 1)

3. How is the width of a strip determined?
   A. Average of the offsets
   B. Difference of the chainages
   C. Fixed 10 m
   D. Sum of the chainages
   Answer: B. Width is the chainage difference between successive offsets. (Key point: Strip width)

4. What is the total area?
   A. 350.00 m2
   B. 412.40 m2
   C. 396.80 m2
   D. 300.00 m2
   Answer: C. 41.4 + 58 + 43.5 + 90 + 80.4 + 83.5 = 396.8 m2, confirmed by coordinates. (Key point: Total area)

## Related in Surveying I

- [Area from offsets, 10 m interval](https://www.bctengineer.com/question.md?key=semester_3%2Fsurveying_i%3Aarea-offsets-4-6-7-2-9-6-6-4) — Surveying I, chapter 9
- [Area from twelve offsets at 20 m interval](https://www.bctengineer.com/question.md?key=semester_3%2Fsurveying_i%3Aarea-offsets-twelve-offsets-4-19) — Surveying I, chapter 9, asked 2×
- [Area from eight offsets at 20 m interval](https://www.bctengineer.com/question.md?key=semester_3%2Fsurveying_i%3Aarea-offsets-20m-interval-eight-offsets) — Surveying I, chapter 9, asked 2×
- [Area from perpendicular offsets to a hedge](https://www.bctengineer.com/question.md?key=semester_3%2Fsurveying_i%3Aarea-offsets-hedge-5-5-12-7) — Surveying I, chapter 9
- [Area by coordinate method from offsets](https://www.bctengineer.com/question.md?key=semester_3%2Fsurveying_i%3Aarea-coordinate-method-chainage-3-75) — Surveying I, chapter 9
- [Offsets area: trapezoidal, Simpson, coordinate](https://www.bctengineer.com/question.md?key=semester_3%2Fsurveying_i%3Aarea-offsets-curved-boundary-trapezoidal-simpson-coordinate) — Surveying I, chapter 9

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