H. Area
1. Traverse with straight sides
a. Closed polygon
The area of any closed noncrossing polygon, Figure H1,
Figure H1 
can be computed using the coordinates of its verticies with Equation H1:
Equation H1 
This equation works for any polygon with straight sides. The more verticies, the more terms in the equation. An easy way to remember equations is graphically:
Starting at one point, list the coordinates in sequence around the exterior. Repeat the first point at the end. 

Crossmultiply the coordinates. 

Sum the crossproducts.


Subtract one sum from the other, divide the result by two, and take the absolute value. This is the polygon area. 

It doesn't matter:
 at which point you start
 going clockwise or counterclockwise around the polygon
 whether coordinates are EastNorth or NorthEast
The last two can affect the area's mathematical sign which is why Equation H2 uses the absolute value.
b. Example area computation
Determine the area of the traverse in Figure H2.
Figure H2 
Set up the coordinates table with additional rows for first point repetition and sums and two columns for crossproducts.
We'll start at point C, travel clockwise, and carry an extra significant figure to minimize cumulative rounding.
Point  North (ft)  East (ft)  
C  406.31  1259.97  
D  235.12  1489.47  
E  65.81  1126.40  
A  317.89  942.04  
B  675.32  1282.54  
C  406.31  1259.97  
sums: 
Partial crossproducts:
406.31 x 1489.47 = 605,187
235.12 x 1126.40 = 264,489
...
235.12 x 1259.97 = 296,244
65.81 x 1489.47 = 98,022
...
The units on the crossproducts are square feet.
Completed table:
Point  North (ft)  East (ft)  
C  406.31  1259.97  296,244  
D  235.12  1489.47  605,187  98,022 
E  65.81  1126.40  264,839  358,071 
A  317.89  942.04  61,996  636,178 
B  675.32  1282.54  407,707  521,109 
C  406.31  1259.97  850,883  
sums:  2,190,612  1,909,624 
Since we carried an extra significant figure, Area = 140,490 sq ft.
c. Noncrossing traverses only
Equation H1 will not return a correct area if a traverse crosses itself. The traverse in Figure H3 represents the order in which the points were surveyed, traverse adjusted, etc. Applying Equation H1 to the coordinates in their surveyed order results in an "area" of 8,412 sq ft. The area is nonsensical since the traverse doesn't have an "inside" like a noncrossing polygon.
Coordinates
Area ABCDA = 8,412 sq ft. 

Figure H3 
If we reorder the point list to a noncrossing perimeter, Figure H4, the area is 70,717 sq ft.
Coordinates
Area ACDBA = 70,717 sq ft. 

Figure H4 
Apply Equation H1 only to a noncrossing traverse with the coordinates listed in order around the perimeter of the desired area.
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).