---
title: "Area by coordinates and DMD (AB -370)"
subject: "Surveying I"
chapter: 9
key: "semester_3/surveying_i:area-coordinate-and-dmd-minus-370"
source: https://www.bctengineer.com/q/semester_3/surveying_i/area-coordinate-and-dmd-minus-370
site: BCT Engineering
---

# Area by coordinates and DMD (AB -370)

Surveying I · Chapter 9: Computation of Area and Volume

- Asked in: 2079 Bhadra, 3+3 marks
- Video: [How to calculate Area of Closed Traverse by Independent Coordinates? | Surveying | Civil Engineering](https://www.youtube.com/watch?v=VfJmks_ox78) — Salim K Tutorials
- Answer checked by Claude Opus 5.5 (high effort) on 2026-10-14
- Page: https://www.bctengineer.com/q/semester_3/surveying_i/area-coordinate-and-dmd-minus-370
- Whole chapter: https://www.bctengineer.com/md/semester_3/surveying_i/chapter-9

## Question

Calculate the area of the traverse by co-ordinate and double meridian distance method.

| Line | AB | BC | CD | DA |
|---|---|---|---|---|
| Latitude (m) | -370 | 240 | -260 | 390 |
| Departure (m) | 220 | -400 | -300 | 480 |

## Short answer

By DMD method, DMDs are 220, 40, −660, −480 m; Σ(L×DMD) = −87400 m², area = 43700 m² (4.37 ha). The coordinate method gives the same result.

## Answer

The traverse closes: ΣL = −370 + 240 − 260 + 390 = 0 and ΣD = 220 − 400 − 300 + 480 = 0.

#### DMD method

**DMD rule (taking the meridian through the starting station):**
- DMD of the first line = departure of the first line.
- DMD of any line = DMD of the previous line + departure of the previous line + departure of the line itself.
- DMD of the last line = departure of the last line with opposite sign (a check).
- Double area = Σ (latitude × DMD), and area = ½ |Σ L × DMD|.

- DMD(AB) = D(AB) = 220
- DMD(BC) = 220 + (220) + (-400) = 40
- DMD(CD) = 40 + (-400) + (-300) = -660
- DMD(DA) = -660 + (-300) + (480) = -480
- Check: DMD(DA) = -480 = −D(DA) = -480, so the DMDs are correct.

| Line | Latitude L (m) | Departure D (m) | DMD (m) | L × DMD (m²) |
|---|---|---|---|---|
| AB | -370 | 220 | 220 | -81400 |
| BC | 240 | -400 | 40 | 9600 |
| CD | -260 | -300 | -660 | 171600 |
| DA | 390 | 480 | -480 | -187200 |
| **Sum** | 0 | 0 | | **-87400** |

$$
\text{Area} = \frac{|-87400|}{2} = 43700\ \text{m}^2
$$

#### Coordinate method

Taking the first station as the origin (N = 0, E = 0) and adding the latitudes and departures successively gives the total coordinates. The area is

$$
2A = \sum N_i\,(E_{i+1} - E_{i-1})
$$

| Station | N (m) | E (m) | E(next) − E(previous) | N × (E(next) − E(previous)) |
|---|---|---|---|---|
| A | 0 | 0 | 700 | 0 |
| B | -370 | 220 | -180 | 66600 |
| C | -130 | -180 | -700 | 91000 |
| D | -390 | -480 | 180 | -70200 |
| **Sum** | | | | **87400** |

$$
\text{Area} = \frac{|87400|}{2} = 43700\ \text{m}^2
$$

Both methods agree.

**Answer: Area = 43700.00 m² (4.3700 ha).**

*Note on the data:* when the stations are plotted, side BC crosses side DA (at about N −136.9, E −168.5), so the figure is a crossed quadrilateral made of two loops of about 46231 m² and 2531 m². Both the DMD and coordinate formulas give the difference of the two loops, 43700 m², which is the answer expected in the exam; the total ground actually enclosed would be about 48762 m².

## Check yourself

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

1. How is closure confirmed?
   A. Only departures sum to 0
   B. Latitudes sum to 0 and departures sum to 0
   C. Latitudes equal departures
   D. Sides are in pairs
   Answer: B. Both sums are zero. (Key point: Closure)

2. How is the DMD of a line (other than the first) found?
   A. DMD of previous line + departure of previous line + departure of the line itself
   B. DMD of previous line - departure of the line itself
   C. Latitude of previous line + latitude of the line
   D. Sum of all departures
   Answer: A. Each DMD = previous DMD + previous departure + own departure. (Key point: DMD rule)

3. How is the area obtained from the DMD table?
   A. sum of (departure x DMD)
   B. sum of (latitude x departure)
   C. sum of DMDs divided by 2
   D. 1/2 x |sum of (latitude x DMD)|
   Answer: D. Double area = sum of latitude x DMD, so area is half its absolute value. (Key point: Area from DMD)

4. Which formula is used for the coordinate method?
   A. A = N x E
   B. 2A = sum (N_i + E_i)
   C. 2A = sum N_i (E_(i+1) - E_(i-1))
   D. A = sum L x D
   Answer: C. Each northing is multiplied by the difference of the next and previous eastings; both methods give 43700 m^2. (Key point: Coordinate formula)

## Close variants set in other exams

- [Area by DMD, traverse ABCDE (+218)](https://www.bctengineer.com/question.md?key=semester_3%2Fsurveying_i%3Admd-traverse-ab-218) — Surveying I, chapter 9, asked 4×
- [Area by DMD, traverse PQRS](https://www.bctengineer.com/question.md?key=semester_3%2Fsurveying_i%3Admd-traverse-pqrs-minus-300) — Surveying I, chapter 9, asked 4×
- [Area by DMD, traverse ABCDEA (-5.693)](https://www.bctengineer.com/question.md?key=semester_3%2Fsurveying_i%3Admd-five-sides-lat-minus-5-693) — Surveying I, chapter 9, asked 4×

## Related in Surveying I

- [Area by total coordinates and DMD (AB 90.29)](https://www.bctengineer.com/question.md?key=semester_3%2Fsurveying_i%3Admd-and-coordinates-ab-90-29) — Surveying I, chapter 9
- [Area by DMD, departure AB 405.85](https://www.bctengineer.com/question.md?key=semester_3%2Fsurveying_i%3Admd-traverse-ab-405-85) — Surveying I, chapter 9, asked 2×
- [Area of traverse by DMD and coordinates (N/S/E/W)](https://www.bctengineer.com/question.md?key=semester_3%2Fsurveying_i%3Aarea-closed-traverse-nswe-108) — Surveying I, chapter 9
- [Area by DMD, traverse ABCD (225.28)](https://www.bctengineer.com/question.md?key=semester_3%2Fsurveying_i%3Admd-traverse-ab-225-28) — Surveying I, chapter 9, asked 2×
- [Meridian distance, DMD and area calculation](https://www.bctengineer.com/question.md?key=semester_3%2Fsurveying_i%3Ameridian-distance-and-dmd-area) — Surveying I, chapter 9, asked 2×
- [Local attraction: AB 69°30′ compass traverse](https://www.bctengineer.com/question.md?key=semester_3%2Fsurveying_i%3Alocal-attraction-ab-69-30-reordered) — Surveying I, chapter 4

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