February 8, 2005


Measurement of the Area by Double Meridian Distance Technique

By Dr. Frank J. Collazo


The meridian distance of a traverse line is equal to the length of a line running east to west from the midpoint of the traverse line to a reference meridian.  The reference meridian is the meridian that passes through the most westerly traverse station.  In figure #1 (Plot), the dotted lines indicate the meridian distances of the traverse lines to which they extend from the reference meridians.


The following rules for determining meridian distance are outlined below:


1.      For the initial traverse line in a closed traverse, the meridian distance equals one half of the departure.


2.      For each subsequent traverse line, the meridian distance equals the meridian distance of the preceding course line, plus one half of the departure of the preceding line, plus one half of the departure of the line itself.


If the rules have been followed correctly, the BMD of the last course will be equal to departure of the last course with its sign changed.


The altitude is the latitude of the course, and the average of the bases of the several courses is equal to the perpendicular distance to each course of the meridian.


Methodology (Equipment Required):  One transit with a tripod, Stadia Rod, and Data Collection forms shown in Figure #1.


Measurements: The Transit operator will be assisted by one individual to position the Stadia Rod under the command of the Transit Operator.  The Transit will measure the range and bearing of the points.  It is envisioned that a minimum of ten readings from each side will be taken.  Figure #1 illustrates the necessary data elements that must be recorded during the survey.  Columns #1is the station number, Column #2 is the slant range, Column #3 bearing angle, and Column #4 the elevation angle.  Columns #5 thru #10 are the elements of data calculated  from the recorded measurements.  The analytical results will be calculated before the measurements.  The algorithm has been tested and verified -- it works.


Calculations:  Table 1 illustrates the series of calculations based on the following measurements: Distance, Bearing, and Altitude.  For this project, it is assumed to be in two dimensions excluding the topography and elevation or altitude considerations.  The recorded measurements are used to calculate the latitude, departure, double meridian distance, and double areas.  At the end of the calculations, the DMD of the last course should be the same or approximated to the departure of the last course except a different sign.  If they are not the same, a truncation error must have occurred.  Figure 1 depicts the geometry of the plot and Table 1 illustrates the calculations of the plot.  A test case has been used to define the area in distances and bearing to calculate the latitude and longitude of each course line.  In essence, the table data elements are the basis to determine the acreage of the land.  A table of the liner measure and area measure standards are provided for information only.


The following is a summary of the formulas used to calculate the area of he farm:


St = Slant Distance in yards.

Bo = The azimuth angle to the course line referenced to North.

Latitude = (St)*Sine Bo (Theta).

Departure = (St)* cosine Bo (Theta).

DMD of the first course = Departure of the first course.

DMD (1) = (DMD of the first course) + (departure of the preceding point) + (the departure of the current point) Note 1.

Double Areas = (Latitude) * (DMD) - Multiply the latitude of the course line by the DMD of the course line.  The results will be either positive or negative.


Note 1: The subsequent courses are calculated in the same fashion.


Appendix 1: Standard  Measurement Used In Surveying


Linear Measure:

12 inches (in.) = 1 foot (ft.)

3 feet = 1 yard (yd)

51/2 yards = 1 rod (rd), pole, or perch (161/2 ft.)

40 rods = 1 furlong (fur) = 220 yds = 660 ft.

8 furlongs = 1 statute mile (mi.) = 1,760 yds = 5,280 ft.

3 land miles = 1 league

5,280 feet = 1 statute or land mile

6,076.11549 feet = 1 international nautical mile


Area Measure:

144 square inches = 1 sq ft.

9 square feet = 1 sq yd = 1,296 sq in.

301/4 square yards = 1 sq rd = 2721/4 sq ft.

160 square rods = 1 acre = 4,840 sq yds = 43,560 sq ft.

640 acres = 1 sq mi.

1 mile square = 1 section (of land)

6 miles square = 1 township = 36 sections = 36 sq mi.


Gunter's or Surveyor's Chain Measure:

7.92 inches = 1 link (li)

100 links = 1 chain (ch) = 4 rods = 66 ft.

80 chains = 1 statute mile = 320 rods = 5,280 ft.