--- title: "Area of traverse by DMD and coordinates (N/S/E/W)" subject: "Surveying I" chapter: 9 key: "semester_3/surveying_i:area-closed-traverse-nswe-108" source: https://www.bctengineer.com/q/semester_3/surveying_i/area-closed-traverse-nswe-108 site: BCT Engineering --- # Area of traverse by DMD and coordinates (N/S/E/W) Surveying I · Chapter 9: Computation of Area and Volume - Asked in: 2080 Bhadra, 3+3 marks - Video: [AREA OF CLOSED TRAVERSE: LATITUDE & DEPARTURE/ ERROR OF CLOSURE](https://www.youtube.com/watch?v=TFfu4_ilVyU) — Engineereya's Vlog - Answer checked by Claude Opus 5.5 (high effort) on 2026-10-14 - Page: https://www.bctengineer.com/q/semester_3/surveying_i/area-closed-traverse-nswe-108 - Whole chapter: https://www.bctengineer.com/md/semester_3/surveying_i/chapter-9 ## Question A closed traverse ABCDA is run along the boundaries with the following results. | Side | Latitude N | Latitude S | Departure E | Departure W | |---|---|---|---|---| | AB | 108 | | 4 | | | BC | 15 | | 249 | | | CD | | 123 | 4 | | | DA | 0 | | | 257 | Calculate the area of the traverse by Double Meridian method and coordinate method. ## Short answer DMD method: DMDs derived from departures, area = ½|Σ(L × DMD)| = 29221.50 m² (2.922 ha). Coordinate method confirms. ## Answer Convert the table to signed values (N and E positive, S and W negative): ΣN = 123 = ΣS and ΣE = 257 = ΣW, so the traverse closes. | Side | Latitude (m) | Departure (m) | |---|---|---| | AB | +108 | +4 | | BC | +15 | +249 | | CD | −123 | +4 | | DA | 0 | −257 | #### 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) = 4 - DMD(BC) = 4 + (4) + (249) = 257 - DMD(CD) = 257 + (249) + (4) = 510 - DMD(DA) = 510 + (4) + (-257) = 257 - Check: DMD(DA) = 257 = −D(DA) = 257, so the DMDs are correct. | Line | Latitude L (m) | Departure D (m) | DMD (m) | L × DMD (m²) | |---|---|---|---|---| | AB | 108 | 4 | 4 | 432 | | BC | 15 | 249 | 257 | 3855 | | CD | -123 | 4 | 510 | -62730 | | DA | 0 | -257 | 257 | 0 | | **Sum** | 0 | 0 | | **-58443** | $$ \text{Area} = \frac{|-58443|}{2} = 29221.5\ \text{m}^2 $$ #### 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 | -253 | 0 | | B | 108 | 4 | 253 | 27324 | | C | 123 | 253 | 253 | 31119 | | D | 0 | 257 | -253 | 0 | | **Sum** | | | | **58443** | $$ \text{Area} = \frac{|58443|}{2} = 29221.5\ \text{m}^2 $$ Both methods agree. **Answer: Area = 29221.50 m² (2.922 ha).** ## Check yourself The answer's key points as multiple-choice questions, with the correct option and why. 1. How is the N/S/E/W table converted before computing? A. All values positive B. N and E positive, S and W negative C. N and W positive, S and E negative D. S and E positive, N and W negative Answer: B. Latitudes: N +, S -. Departures: E +, W -. (Key point: Signs) 2. How does the answer show the traverse closes? A. Sum N = sum S = 123 and sum E = sum W = 257 B. Because all sides are equal C. Because DA has zero latitude D. Because there are four sides Answer: A. Closure needs net latitude and net departure to be zero. (Key point: Closure check) 3. How is the DMD of a line (other than the first) found? A. DMD of previous line - departure of the line itself B. Latitude of previous line + latitude of the line C. Sum of all departures D. DMD of previous line + departure of previous line + departure of the line itself Answer: D. Each DMD = previous DMD + previous departure + own departure. (Key point: DMD rule) 4. With DMD(AB) = 4 and departures AB = 4, BC = 249, what is DMD(BC)? A. 253 B. 249 C. 257 D. 4 Answer: C. 4 + 4 + 249 = 257. 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