Area by coordinates and DMD (AB -370)

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.

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

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

$$2A = \sum N_i\,(E_{i+1} - E_{i-1})$$ $$\text{Area} = \frac{|87400|}{2} = 43700\ \text{m}^2$$

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.