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.