Article Index

1. Traverse with straight sides

a. Closed polygon

The area of any closed non-crossing polygon, Figure H-1,

Figure H-1
Closed non-crossing polygons

can be computed using the coordinates of its verticies with Equation H-1:

Equation H-1

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.

 

Cross-multiply the coordinates.

Sum the cross-products.

 

  

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


Equation H-2

It doesn't matter:

  • at which point you start
  • going clockwise or counterclockwise around the polygon
  • whether coordinates are East-North or North-East

The last two can affect the area's mathematical sign which is why Equation H-2 uses the absolute value.

b. Example area computation

Determine the area of the traverse in Figure H-2.

Figure H-2
Area Example

Set up the coordinates table with additional rows for first point repetition and sums and two columns for cross-products.

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 cross-products:


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 cross-products 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. Non-crossing traverses only

Equation H-1 will not return a correct area if a traverse crosses itself. The traverse in Figure H-3 represents the order in which the points were surveyed, traverse adjusted, etc. Applying Equation H-1 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 non-crossing polygon.

Coordinates

Point N (ft) E (ft)
A 1000.00 2000.00
B 1248.80 1881.25
C 1019.65 1607.03
D 1217.88 1643.52

Area ABCDA = 8,412 sq ft.

Figure H-3
Crossing traverse 

 

If we re-order the point list to a non-crossing perimeter, Figure H-4, the area is 70,717 sq ft.

Coordinates

Point N (ft) E (ft)
A 1000.00 2000.00
C 1019.65 1607.03
D 1217.88 1643.52
B 1248.80 1881.25

Area ACDBA = 70,717 sq ft.

Figure H-4
Crossing traverse 

 

Apply Equation H-1 only to a non-crossing traverse with the coordinates listed in order around the perimeter of the desired area.

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