--- title: "Area by total coordinates and DMD (AB 90.29)" subject: "Surveying I" chapter: 9 key: "semester_3/surveying_i:dmd-and-coordinates-ab-90-29" source: https://www.bctengineer.com/q/semester_3/surveying_i/dmd-and-coordinates-ab-90-29 site: BCT Engineering --- # Area by total coordinates and DMD (AB 90.29) Surveying I · Chapter 9: Computation of Area and Volume - Asked in: 2081 Baisakh, 6 marks - Answer checked by Claude Opus 5.5 (high effort) on 2026-10-14 - Page: https://www.bctengineer.com/q/semester_3/surveying_i/dmd-and-coordinates-ab-90-29 - Whole chapter: https://www.bctengineer.com/md/semester_3/surveying_i/chapter-9 ## Question Calculate the area of the traverse by total coordinates and DMD methods. | Line | AB | BC | CD | DA | |---|---|---|---|---| | Latitude (m) | 90.29 | -151.00 | -116.10 | 176.81 | | Departure (m) | 246.39 | 35.00 | -241.55 | -39.84 | ## Short answer Double Meridian Distance method: compute DMDs, area = ½|Σ(L × DMD)| = 43849.46 m² (4.3849 ha). The coordinate method gives the same result. ## Answer The traverse is closed (ΣL = 0.00 and ΣD = 0.00), so the area can be found directly from the latitudes and departures. #### 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) = 246.39 - DMD(BC) = 246.39 + (246.39) + (35.00) = 527.78 - DMD(CD) = 527.78 + (35.00) + (-241.55) = 321.23 - DMD(DA) = 321.23 + (-241.55) + (-39.84) = 39.84 - Check: DMD(DA) = 39.84 = −D(DA) = 39.84, so the DMDs are correct. | Line | Latitude L (m) | Departure D (m) | DMD (m) | L × DMD (m²) | |---|---|---|---|---| | AB | 90.29 | 246.39 | 246.39 | 22246.55 | | BC | -151.00 | 35.00 | 527.78 | -79694.78 | | CD | -116.10 | -241.55 | 321.23 | -37294.80 | | DA | 176.81 | -39.84 | 39.84 | 7044.11 | | **Sum** | 0.00 | 0.00 | | **-87698.92** | $$ \text{Area} = \frac{|-87698.92|}{2} = 43849.46\ \text{m}^2 $$ #### Coordinate method Taking the first station as the origin (N = 0, E = 0) and adding 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 | 0.00 | 0.00 | 206.55 | 0.00 | | B | 90.29 | 246.39 | 281.39 | 25406.70 | | C | -60.71 | 281.39 | -206.55 | 12539.65 | | D | -176.81 | 39.84 | -281.39 | 49752.57 | | **Sum** | | | | **87698.92** | $$ \text{Area} = \frac{|87698.92|}{2} = 43849.46\ \text{m}^2 $$ Both methods agree. **Answer: Area = 43849.46 m² (4.3849 ha).** ## Check yourself The answer's key points as multiple-choice questions, with the correct option and why. 1. Why can the area be computed directly from the given latitudes and departures? A. The traverse has four sides B. The traverse closes: sum of latitudes and sum of departures are both zero C. The angles sum to 360 degrees D. The lengths are equal Answer: B. A closed traverse has zero sum of latitudes and departures, so no correction is needed. (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. What check is available on the DMDs? A. The sum of DMDs is zero B. The first and last DMDs are equal C. The last DMD equals the last latitude D. The DMD of the last line equals its departure with opposite sign Answer: D. With the first-line DMD starting at its departure, the last DMD must equal minus the last departure. (Key point: DMD check) 4. What does the coordinate method give compared with the DMD method? A. A different area, to be averaged B. Half the area C. Same area, 43849.46 m^2 D. Twice the area Answer: C. Both methods give 43849.46 m^2 (4.3849 ha), which confirms the work. 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