Summary
An ever-increasing number of farmers have global positioning system (GPS) receivers on their combines. When
not harvesting, GPS receivers are useful for more than locating one’s favorite fishing spot. They can be tools for
determining the distance between two points or to accurately determine the acres in a field that is to be rented. The
distance between two sampling points and the area of a field can be found using GPS coordinates and knowledge of
the Earth Terrestrial Coordinate System. Because GPS latitude and longitude are in terrestrial coordinates, determin-
ing the distance in length measurements (feet, meters, yards, kilometers, and miles) rather than degrees, between two
points is not trivial. The objective of this guideline is to provide a method that farmers, ranchers, or agricultural
practitioners can use to calculate distances between points and to calculate the size of a field using Excel, a commonly
available spreadsheet. A more detailed description for calculating distance and area is found in Carlson (1999), in which
the mathematics and assumptions used to create the model used in this guideline are described. A Basic programming
language computer program is also available in that paper.
The Site-Specific Management Guidelines series is published by the Potash & Phosphate Institute (PPI) • Coordinated by South Dakota State University (SDSU)
Sponsored by the United Soybean Board (USB) and the Foundation for Agronomic Research (FAR). For more information, call (605) 692-6280. www.ppi-far.org/ssmg
C.G. Carlson and D.E. Clay SSMG-11
Simplifying Assumptions
For purposes of these calculations we will go back to
the middle ages and assume that the earth is flat. On a flat
earth, a line of longitude is parallel to the next line of
longitude and lines of longitude are perpendicular to lines
of latitude. After study of Figure 1, it is clear that these
assumptions are not true. However, calculations show that
if distances are small (less than one mile on the earth
surface), these assumptions result in errors of less than 16
inches in a mile at 45 degrees latitude. Errors are smaller
close to the equator and larger as the poles are ap-
proached. Also important to note is the fact that there
have been many analytical models to describe the earth.
Most GPS receivers and our calculations use the WGS84
spheroid earth model. Constants for the major and minor
axis for this model are 6,378,137.0 meters and
6,356,752.3142 meters, respectively.
Distance between Two Points
First, use a DGPS (differential corrected GPS) to
determine the latitude and longitudes of two points on the
earth surface and determine an approximate elevation. In
the example, we have used a Trimble 132 DGPS receiver to
determine the latitude and longitude of the ends of a 300 ft.
tape (Table 1). GPS receivers report locations in both
decimal degrees and degrees, minutes, and seconds. GPS
data reported in degrees, minutes, and seconds must first
be converted to decimal degrees.
Converting is best accomplished by using examples.
Suppose you want to convert -96 degrees, 47.46732
minutes and 44 degrees, 27.66786 minutes into decimal
degrees. This is accomplished by dividing the minutes by
60 and adding together as follows:
Decimal degrees = -(96 + (47.46732/60)) = -96.791122
44 + (27.66786/60) = 44.46113
The Earth Model—Calculating Field Size and
Distances between Points using GPS Coordinates
Figure 1. Latitude and longitude of the Earth.
Distance between two adjacent integer
(whole numbers such as 1 and 2, or 25 and
26, not 13.3 and 13.22 degrees) longitude
lines are approximately 70 miles at the
equator and converge to 0 miles at the
North or South Poles.
2
Table 2a. Input, equations, and output of an Excel spreadsheet. Once the equations are input, distances between
different points can be calculated by changing B1, B2, C1, C2, and C15.
ABC
1 Point #1 Latitude point 1 (45.4649135) Longitude point 1 (-95.9090323)
2 Point #2 Latitude point 2 (45.46555306) Longitude point 2 (-95.9097683)
3 Determine true angle =(ATAN((C14^2)/(C13^2)*TAN(B1*PI()/180)))*180/PI()
4 Determine true angle =(ATAN((C14^2)/(C13^2)*TAN(B2*PI()/180)))*180/PI()
5
6 Radius pt 1 =(1/((COS(B3*PI()/180))^2/C13^2+(SIN(B3*PI()/180))^
2/C14^2))^0.5+C15
7 Radius pt 2 =(1/((COS(B4*PI()/180))^2/C13^2+(SIN(B4*PI()/180))^
2/C14^2))^0.5+C15
8
9 X - Y earth coordinates =B6*COS(B3*PI()/180) =B6*SIN(B3*PI()/180)
10 X - Y earth coordinates =B7*COS(B4*PI()/180) =B7*SIN(B4*PI()/180)
11
12 X coordinate =((B9-B10)^2+(C9-C10)^2)^0.5
13 Y coordinate =2*PI()*((((B9+B10)/2))/360)*(C1-C2) 6378137
14 6356752.3142
15 Distance meter =((B12)^2+(B13)^2)^0.5 Elevation meters (334.9)
16 Distance feet = B15*3.28084
17
18
19
20 Area of a triangle =0.5*ABS(B17*C18-C17*B18+C17*D18-D17*C18+D17*
B18-B17*D18)
21 M^2 to ft^2 =(B20*10.76391)
22 Ft^2 to acres =(B21/43560)
The same data may have been presented as -96 degrees,
47 minutes, and 28.0392 seconds and 44 degrees, 27
minutes, and 40.0716 seconds. Use the same procedure as
above, but divide the seconds by 60*60 or 3600. The
results are in units of decimal degrees.
Decimal degrees = -(96 + (47 /60) + (28.0392/3600)) = -
96.791122
Second, input the latitude and longitude information
and appropriate equations into an Excel worksheet (Table
2a).
Table 1. Latitude, longitude, and elevations of the two
ends of a 300 ft. tape.
Latitude Longitude Elevation
Point 1 45.46491350 -95.90903233* 334.9 m
Point 2 45.46555306 -95.90976827 334.9 m
* Note that in the Western Hemisphere, to the west of Greenwich,
England, longitude is many times expressed as a negative number.
This distinguishes a latitude-longitude point from the same point in
the Eastern Hemisphere. It also simplifies the mathematics of
calculating distance. In the Southern Hemisphere, latitude becomes
negative for the same reason.
When the equations and constants are input correctly,
the Excel spreadsheet will have the values shown in Table
2b.
The distance between the two points is determined by
converting the latitude/longitude values to values on an
X/Y coordinate plane. Point 1 is at the origin (0,0), and
point 2 is at the coordinates shown in B12 and B13
(71.08488298, 57.55806184). The distance between the
points is solved using the equation:
Distance = ((0-71.08)
2
+(0-57.6)
2
))
0.5
.
This value is in B15. The distance 91.465877 m can be
converted to feet (300.0849 ft) by multiplying it by the
conversion factor 3.28084 ft/m.
Calculation of Field Area
A second exercise is presented to give an example
calculation of the area of a field. Again, the same simpli-
fying assumptions (flat and no curvature of the longitudi-
nal lines) are appropriate for calculation of an area that is
small relative to the size of the earth…less than 1 section
(640 acres).
3
Figure 2. Practice football field divided into calculable
triangles.
Select the smallest value in F1 through 4
(F2=44.3210783) and G 1 through 4 (G1= -96.7779358)
and put these values in B2 and C2. These values will be
defined as the origin on the X/Y coordinate system and
the points are located in the southwest corner of the area
being worked on.
The area in triangle 1 will be calculated in steps 3
through 6.
Step 3: Copy F1 and G1 to B1 and C1, respectively.
The X and Y coordinates for point 1 are in B12 and
B13. Copy these points to B17 and B18. When
copying these values you must copy the values, not
the equation. This can be accomplished by using
{paste special} and {values} commands.
Step 4: Copy F2 and G2 to B1 and C1, respectively.
The X and Y coordinates for point 2 are in B12 and
B13; copy these values to C17 and C18. When
copying these values you again must copy the value,
not the equation.
Step 5: Copy F3 and G3 to B1 and C1, respectively.
The X and Y coordinates for point 3 are in B12 and
B13. Copy these values to D17 and D18. When
copying these values you must copy the value, not
the equation.
When you are done the values in the spreadsheet
should be:
BC D
17 49.55035245 0 48.45022
18 0 0.207419129 91.6629606
Step 6: calculate the area for triangle 1 using the
formula given above. This is accomplished by
setting cell:
B20 = 0.5*ABS(B17*C18-C17*B18+C17*D18-
D17*C18+D17*B18-B17*D18)
The value (2270.852m
2
) in cell B20 represents the
area of triangle 1. This value is converted to ft
2
by
multiplying it by 10.76391ft
2
/m
2
. The 24,443.25 ft
2
is then converted to acres by dividing it by 43,560
ft
2
/acre. Following these calculations the area of
triangle 1 is 0.56114 acres.
Step 7: Repeat steps 2 through 5 for triangle 2. As
defined under step 2, your new corner points are
located at points ii(F2,G2), iii(F3,G3), and
iv(F4,G4). Calculations are accomplished by
Table 2b. The values in the Excel spread sheet
following input of values and equations
shown in Table 2a.
ABC
1 Point #1 45.4649135 -95.90903233
2 Point #2 45.46555306 -95.90976827
3 Determine true angle 45.27250514
4 Determine true angle 45.27314475
5
6 Radius pt 1 6367650.922
7 Radius pt 2 6367650.683
8
9 X - Y earth coordinates 4481143.389 4523973.05
10 X - Y earth coordinates 4481092.718 4524022.91
11
12 X coordinate 71.08499298
13 Y coordinate 57.55806184 6378137
14 6356752.31
15 Distance meters 91.46587729 334.9
16 Distance feet 300.0849088
For our example, the four corners of a football practice
field at South Dakota State University were located using
a Trimble 132 receiver.
Calculating Areas of a Field
In calculating areas, the simplifying assumptions
discussed above are used. In our example you will need
to: (i) use the spread sheet developed above; (ii) know the
DGPS coordinates of the corners of the field in question;
and (iii) use the triangle area equation, which is:
Area = .5*|x
1
*y
2
- x
2
*y
1
+ x
2
*y
3
- x
3
*y
2
+ x
3
*y
1
- x
1
*y
3
|
Note that the | | lines are absolute value operators.
This means that the calculation inside of the | | will
be positive or converted to positive.
where x
i
and y
i
are the coordinates of the three points
making the triangle.
To calculate acres:
Step 1: Use a DGPS to measure the corners of your test
field. The latitude and longitude values of South
Dakota State University football field are given in
step 2
Step 2: Separate the field into two triangles (or as many
triangles as are necessary to cover the entire field.
Irregular fields can be approximated by a number of
triangles). This is accomplished by identifying points
i, ii, and iii as the corners for triangle 1 and the points
ii, iii, and iv as the corners for triangle 2.
Input the longitude and latitude information into the F
and G columns. When done, the spreadsheet should look
like:
FG
1 44.3215242 -96.7779358
2 44.3210783 -96.7779332
3 44.3215143 -96.7767868
4 44.321080 -96.7767883
4
copying F2 and G2 to B1 and C1; F3 and G3 to B1
and C1; and F4 and G4 to B1 and C1. When done
correctly the spreadsheet will have the values:
BC D
17 48.450221 0.188911406
18 0.2074192 91.6629606 91.62341041
19
20 2203.991
Following the calculations described above the
number of acres contained in triangle 2 is 0.54462
Step 8. Determine total acres by adding the acres in 6
and 7 together. The total acres in the football field
were 1.10576. ■
Ref. # 99081 / Item # 10-1011
This Site-Specific Management Guideline was prepared by:
Dr. C. Gregg Carlson
Professor, Plant Science
South Dakota State University
208 Ag Hall, P.O. Box 2207A
Brookings, SD 57007
Phone: (605) 688-4761
E-mail: [email protected]
Dr. David Clay
Associate Professor, Soil Science
South Dakota State University
214 Ag Hall, P.O. Box 2207A
Brookings, SD 57007
Phone: (605) 688-5081
E-mail: clayd@ur.sdstate.edu
References
Carlson, C.G. 1999. What do latitude and longitude readings from a DGPS
receiver mean? http://www.abs.sdstate.edu/plantsci/ext/pawg/
earthmo1.htm. C.G. Carlson, Plant Science Department, South Dakota
State University, Brookings, SD
The mention of a commercial name or product does not imply
endorsement of the product named or company named by the
authors, the Plant Science Department, the South Dakota
Cooperative Extension Service, The South Dakota Agricultural
Experiment Station, the College of Agricultural and Biological
Science, or South Dakota State University.
