2. Area By Coordinates
a. Concept
The area of a closed noncrossing plane polygon can be computed from the coordinates of the polygon's verticies. The area equations, depending on coordinate format used, are shown here as Equations G1 and G2
In terms of X and Y coordinates 



Equation G1 
In terms of N and E coordinates 

Equation G2 
These equations look complex, more so if you expand all the terms. Rather than memorize each equation, it's easier to remember their pattern and determine the area in tabular fashion.
To accurately compute the traverse area:
(1) Select a start point (it doesn't matter where you start)
(2) Going in sequence around the exterior list each coordinate pair vertically, Figure G5. Direction of travel around the traverse: clockwise or counterclockwise doesn't matter, nor does coordinate precedence, eg, (X, Y) or (Y, X).
Using X,Y coords 
Using N,E coords 
Figure G5 Listing Coordinates in Sequence 
(3) The first coordinate pair must be repeated at the bottom of the list. In Figure G5, the surveyor started at point A and then ended on point A.
(4) Cross multiply the coordinates and sum the products, Figure G6. Arrows indicate "direction" of multiplication.
Using X,Y coords 
Using N,E coords 
Figure G6 Coordinate Cross Multiplication 
The units of the cross products are square linear units  if coordinates are in feet, cross products are sq ft.
(5) Using Equation G3, compute the traverse area.
Sum each column () and ()
Equation G3 
The absolute value is used since the area could be negative depending on the combination of direction around the traverse, coordinate precedence (e.g. X,Y vs Y,X), and cross multiplication order. Just as the square root of 4 can be either +2 or 2, so can the area be positive or negative. Because we're generally interested in the magnitude of the answer, we use the absolute value of the area computed.
While at first all this may look confusing, it's actually pretty easy to remember once you do it a few times.
b. Examples
Although we carried an additional digit in all previous computations, we'll carry a few more extra here. We'll discuss the area error at the end of the chapter but for now we want to overcompute the area accuracy then report it to an appropriate resolution after we analyze it. If we don't carry additional digits, we could easily increase error due to rounding. To be on the safe side, we'll carry the computations to 0.1 which should be less than the expected error.
(1) Example 1
Figure G7 is a continuation of the Bearing Traverse example we have been using in the past few chapters.


Figure G7 Example 1 Traverse 
Step (1) Start at point A and going clockwise around the traverse list the coordinates:
Point 
N (ft) 
E (ft) 

A 
500.000 
2000.000 

B 
323.614 
1561.426 

C 
526.996 
1488.321 

D 
719.336 
1686.956 

A 
500.000 
2000.000 
remember to return to A 
Step (2) Cross multiply in one direction:
Point 
N (ft) 
E (ft) 
(), sq ft 

A 
500.000 
2000.000 
647,228.0 

B 
323.614 
1561.426 
822,865.2 

C 
526.996 
1488.321 
1,070,602.9 

D 
719.336 
1686.956 
843,478.0 

A 
500.000 
2000.000 







Step (3) Cross multiply in the other direction
Point 
N (ft) 
E (ft) 
(), sq ft 
(), sq ft 
A 
500.000 
2000.000 
647,228.0 

B 
323.614 
1561.426 
822,865.2 
780,713.0 
C 
526.996 
1488.321 
1,070,602.9 
481,641.5 
D 
719.336 
1686.956 
843,478.0 
889,019.1 
A 
500.000 
2000.000 
1,438,672.0 
Step (4) Add up the columns
Point 
N (ft) 
E (ft) 
(), sq ft 
(), sq ft 
A 
500.000 
2000.000 
647,228.0 

B 
323.614 
1561.426 
822,865.2 
780,713.0 
C 
526.996 
1488.321 
1,070,602.9 
481,641.5 
D 
719.336 
1686.956 
843,478.0 
889,019.1 
A 
500.000 
2000.000 
1,438,672.0 



sums: 
3,384,174.1 
3,590,045.6 
Step (5) Using Equation G3 compute the area
Until we discuss area accuracy more fully, we'll state the area as 102,935.8 sq ft.
There's nothing magical or sacred about point A. We could have stated our list at point C and traveled counterclockwise around the traverse. As long as we remember to repeat the initial point at the bottom of the list, we will come up with the same area although one could be positive and the other negative.
(2) Example 2
Figure G8 shows the Crossing Traverse we've been using as another running example.


Figure G8 Crossing Traverse 
With a crossing traverse, one must be careful when listing the coordinates. In this case, if you list the coordinates along the original traverse path, EFGHE, you will be able to compute an area but it will be nonsensical. The traverse turns itself inside out.
Recall that this survey was on a foursided parcel having two obstructed lines, Figure G9.
Figure G9 
We want the the area of the parcel, not the traverse.
Step (1) Start at point E and going clockwise around the parcel, list the coordinates.
Point 
N (ft) 
E (ft) 

E 
1000.000 
200.000 

G 
896.890 
627.584 

F 
689.206 
532.694 

H 
692.474 
257.460 

E 
1000.000 
200.000 
remember to return to E  
Step (2)(4) Cross multiply in both directions; sum the columns.
Point 
N (ft) 
E (ft) 
(), sq ft 
(), sq ft 
E 
1000.000 
200.000 
179,378.0 

G 
896.890 
627.584 
432,534.7 
624,584.0 
F 
689.206 
532.694 
368,876.7 
477,767.9 
H 
692.474 
257.460 
257,460.0 
177,443.0 
E 
1000.000 
200.000 
138,494.8 




1,238,252.4 
1,418,289.7 
Step (5) Use Equation (G3) to compute the area