Why can the area be computed directly from the given latitudes and departures?

The traverse closes: sum of latitudes and sum of departures are both zero

How is the DMD of a line (other than the first) found?

DMD of previous line + departure of previous line + departure of the line itself

DMD of previous line - departure of the line itself

Latitude of previous line + latitude of the line

The DMD of the last line equals its departure with opposite sign

The first and last DMDs are equal

The last DMD equals the last latitude

A different area, to be averaged

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.

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

The traverse is closed (ΣL = 0.00 and ΣD = 0.00), so the area can be found directly from the latitudes and departures.

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

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).

✓ Answer checked by Claude Opus 5.5 (high effort)