---
title: "Area from perpendicular offsets to a hedge"
subject: "Surveying I"
chapter: 9
key: "semester_3/surveying_i:area-offsets-hedge-5-5-12-7"
source: https://www.bctengineer.com/q/semester_3/surveying_i/area-offsets-hedge-5-5-12-7
site: BCT Engineering
---

# Area from perpendicular offsets to a hedge

Surveying I · Chapter 9: Computation of Area and Volume

- Asked in: 2078 Bhadra, 5 marks
- Video: [Calculation of Area by Simpson's, Trapezoidal & mid ordinate method | Type:5 | Surveying 1st |](https://www.youtube.com/watch?v=8Kclh1Fu8R4) — Prashant YT
- 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-hedge-5-5-12-7
- Whole chapter: https://www.bctengineer.com/md/semester_3/surveying_i/chapter-9

## Question

The following perpendicular offsets were taken from a chain line to a hedge.

| Chainage (m) | 0 | 5.5 | 12.7 | 25.5 | 40.5 |
|---|---|---|---|---|---|
| Offset (m) | 5.25 | 6.5 | 4.7 | 5.2 | 4.2 |

Find the area enclosed by the boundary, using any two methods.

## Short answer

Applying the trapezoidal strip-by-strip method to the unequal intervals gives 206.49 m². The coordinate method, treating connections as straight lines, confirms 206.49 m².

## Answer

The offsets are at **unequal** intervals (5.5, 7.2, 12.8 and 15.0 m), so Simpson's rule cannot be used directly. Two methods that work for unequal intervals are the trapezoidal (strip) method and the coordinate method.

#### Method 1: Trapezoidal rule (strip by strip)
Each strip between two successive offsets is a trapezium with area $w(o_1 + o_2)/2$.

| Strip | Width $w$ (m) | $o_1$ (m) | $o_2$ (m) | Area $= w(o_1+o_2)/2$ (m²) |
|---|---|---|---|---|
| 1 | 5.50 | 5.25 | 6.50 | 32.3125 |
| 2 | 7.20 | 6.50 | 4.70 | 40.3200 |
| 3 | 12.80 | 4.70 | 5.20 | 63.3600 |
| 4 | 15.00 | 5.20 | 4.20 | 70.5000 |
| **Total** | | | | **206.4925** |

Area by trapezoidal rule = **206.493 m²**.

#### Method 2: Coordinate method
Chain line is the x-axis; the points in order round the figure are:

| Point | $x$ (m) | $y$ (m) |
|---|---|---|
| Chain start | 0.00 | 0.00 |
| Chain end | 40.50 | 0.00 |
| offset 5 | 40.50 | 4.20 |
| offset 4 | 25.50 | 5.20 |
| offset 3 | 12.70 | 4.70 |
| offset 2 | 5.50 | 6.50 |
| offset 1 | 0.00 | 5.25 |

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

Area = **206.493 m²**, which agrees with the first method.

**Answer:** area enclosed between the chain line, the hedge and the first and last offsets = **206.49 m²** (by both methods).

## Check yourself

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

1. Why can Simpson's rule not be used directly?
   A. There are too few offsets
   B. The offsets are too large
   C. Hedges are curved
   D. The offsets are at unequal intervals
   Answer: D. Simpson's rule needs equal spacing; strip trapezoids work for any widths. (Key point: Unequal intervals)

2. What is the area of a strip between two offsets?
   A. w (o1 + o2)/2
   B. w x o1 x o2
   C. w (o1 - o2)
   D. (o1 + o2)/w
   Answer: A. Each strip is a trapezium of width w and parallel sides o1 and o2. (Key point: Strip formula)

3. What is the area of strip 1 (width 5.5, offsets 5.25 and 6.5)?
   A. 28.88 m2
   B. 32.31 m2
   C. 35.75 m2
   D. 64.63 m2
   Answer: B. 5.5 x 11.75/2 = 32.3125 m2. (Key point: First strip)

4. What is the total area?
   A. 186.40 m2
   B. 212.00 m2
   C. 206.49 m2
   D. 196.50 m2
   Answer: C. 32.3125 + 40.32 + 63.36 + 70.50 = 206.4925 m2. (Key point: Total)

## Related in Surveying I

- [Area enclosed by chain line and hedge](https://www.bctengineer.com/question.md?key=semester_3%2Fsurveying_i%3Aarea-offsets-hedge-5m-interval-simpson-trapezoidal) — Surveying I, chapter 9
- [Area between boundary, offsets and base line](https://www.bctengineer.com/question.md?key=semester_3%2Fsurveying_i%3Aarea-offsets-chainage-0-12-20-25-34-42-52) — 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
- [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 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×