2. Traverse with curves
a. Segments and sectors
Area by coordinates is constrained to following straight lines. Where curves are involved, Figure H3(a), the method returns an area bounded by their chords, Figure H3(b).


Figure H3 
To obtain the correct area, sector or segment areas are computed and added or subtracted accordingly. For the traverse in Figure H3(a), the two segment areas are computed, Figure H4(a), and one subtracted from, the other added to, the area by coordinates, Figure H4(b).
(a) Segments 
(b) A_{Total}=A_{1}A_{2}+A_{3} 
Figure H4 
b. Example area computation
Given the traverse in Figure H5, determine its area.
Figure H5 
In the table below, the points and their coordinates are in the first three columns; their crossproducts are in the last two.
Point  North (ft)  East (ft)  
1  500.00  1200.00  737,634  
2  614.70  1249.70  624,850.  843,243 
3  674.76  1347.43  828,261  892,620. 
4  662.46  1552.76  1,047,730  725,778 
5  467.41  1516.87  1,004,860  633,075 
6  417.36  1317.39  615,766  658,696 
1  500.00  1200.00  500,828  
sums:  4,491,046  4,622,300. 
The area is
This is the area bounded by the chords. We must account for the segments.
Recall that the area of a segment, bounded by an arc and its chord, is
Equation H2 
Compute the area of each segment
Arc 23
Arc 45
Arc 61
Add or subtract the segments:
Total traverse area is 68,790 sq ft.
c. Segment or sector?
Consider the curvinlinear traverse in Figure H5:
Figure H5 
Its area can be computed using the arc sector, Figure H6,
(a) Area to radial lines, A_{R} 
(b) Sector area, A_{Sec} 
Figure H6 Area = A_{R} + A_{Sec} 
or using the arc segment, Figure H7
(a) Area to chord, A_{C} 
(b) Segment area, A_{Seg} 
Figure H7 
Either method is fine. The only difference is that sectors require computing arc radius points, Figure H6(a); segments do not, Figure H7(a).