Finding the Size of the Earth using GPS
Around 240 BC Eratosthenes found the circumference of the Earth by observing shadow
from high noon sun in the cities of Alexandria and Syene. His final value for the
circumference of the Earth was within 25% of the value we used today.1 Eratosthenes
measured the circumference of the earth without leaving Egypt.
Eratosthenes learned that in a town called Syene, now Aswan, along the Nile River at the
summer solstice the Sun at noon shown straight down a deep well, casting no shadow on
the walls. Eratosthenes also knew that Syene was on the Tropic of Cancer and that the
Sun would be directly overhead.2 He also knew, from using a vertical stick and
measuring the cast shadow, that in his hometown of Alexandria, the angle of elevation of
the Sun would be 83° or 7° south of the zenith at the same time. Assuming that
Alexandria was due north of Syene,3 he concluded, using geometry of parallel lines, that
the distance from Alexandria to Syene must be 7/360 of the total circumference of the
Earth. The distance between the cities was well known from travelling caravan to be
about 5,000 stadia. He established a final value of 700 stadia per degree, which implies a
circumference of 252,000 stadia. The exact size of the stadion he used is no longer
known (the common Attic stadion was about 185 m), but it is generally believed that
Eratosthenes' value corresponds to between 39,690 km and 46,620 km. The
circumference of the Earth around the poles is now measured at around 40,008 km.
Eratosthenes result is not bad at all.
Eratosthenes' experiment was highly regarded at the time, and his estimate of the Earth’s
size was accepted for hundreds of years afterwards.
In a sense, Eratosthenes found the size of the earth by slicing a section of the earth as a
piece of pie. Then finding what faction his slice of the pie was when compared to the
whole pie of the earth, he could then calculate the size of the earth. Eratosthenes’ pie
slice from Syene to Alexandria was about 1/50th of the pie. Thus multiply the arc length
of the piecrust by 50 he found a value for the size of the earth.
1. This depends on which distance standard Eratosthenes used.
2. Syene was in fact slightly north of the tropic.)
3. Alexandria is in fact on a more westerly longitude.
We are going to do the same thing using a GPS unit and some math.
Equipment
GPS Capable Device * (We are using a Garmin Etrex Legend)
*The GPS Device needs to be able to display coordinates in Degree-Minutes decimal (DDD°MM.MMM).
For example a manhole in Iowa City might have these coordinates: Latitude: N41°39.059 Longitude:
W091°29.225
Pencil
Calculator
Tape Measures, or meter sticks, distance wheel
Paper
Procedure
For Eratosthenes Alexandria and Syene were for the most part directly north and south of
each other. This greatly simplifies thing. We are going to do the same thing.
The blacktop parking lot located just outside of Strahan Hall has borders that run north,
south, east and west. By walking from the southeast (A) corner of the parking to the
northeast (B) corner we will travel due north in a relatively straight line. Using the GPS
unit we will record two sets of number the latitude (and longitude) of each corner and the
distance between them.
Turn on the GPS unit, or start the GPS function on your device. Your device may need a
few minutes to orient it's self and be ready to navigate.
Starting in the Southeast corner of the main parking lot. This location is the closest
corner to campus map sign in front of Strahan Hall. Record the Longitude and Latitude
for this location.
Southeast Corner Coordinates (Point A)
Latitude (DDD°MM.mmm) :
N____________________________
Longitude (DDD°MM.mmm) :
W____________________________
Walk due north toward the northeast corner of the main parking lot. When you reach the
northeast corner stop and record the longitude and latitude of that location.
Northeast Corner Coordinates (Point B)
Latitude (DDD°MM.mmm) :
N____________________________
Longitude (DDD°MM.mmm) :
W____________________________
Using the distance wheel, tape measure or meter stick measure the distance between the
southeast corner and the northeast corner of the parking lot to the near centimeter. In
other word find the distance in meter to two decimal places. Record the distance.
Record the distance between the southwest corner and northwest corner of the parking lot
in meters.
____________________________m
We have another north-south line we can use as well. Move to the southwest corner.
Perform the same set of measurements by walking in between the southwest and
northwest corners of the parking lot. Record your values below.
Southwest Corner Coordinates (Point C)
Latitude (DDD°MM.mmm) :
N____________________________
Longitude (DDD°MM.mmm) :
W____________________________
Walk due north toward the northeast corner of the main parking lot. When you reach the
northeast corner stop and record the longitude and latitude of that location.
Northwest Corner Coordinates (Point D)
Latitude (DDD°MM.mmm) :
N____________________________
Longitude (DDD°MM.mmm) :
W____________________________
Record the distance between the southwest corner and northwest corner of the parking lot
in meters.
____________________________m
Computation and Data Analysis
Step 1: Write down the latitude for Point A (Southeast Corner):
________________
Step 2: Write down the latitude for Point B (Northeast Corner):
________________
Step 3: Subtract the value of Point A from the value of Point B:
________________
It would be highly unusual to change in the degrees over this short distance. In most
cases you may just subtract the minute decimals from each other. (Point B: MM.mmm –
Point A: MM.mmm)
If each pie slice is the same distance we walked during our experiment, now we need to
know how many “pie slices” are in our earth. Remember Eratosthenes came up with 50
pie slices in his experiment. Our number should be much larger because we walk a much
smaller distance than the distance between Syene and Alexandria. Take the value of the
number of arc minutes in a full circle, 21600” and divide by Step 3 .
Step 4: Divide 21600 by Step 3.
This is number of “pie slices” in our earth.
________________
Step 5: Write down the distance between Point A and Point B.
This value should be in meters.
_______________m
Step 6: Multiply Step 4 by Step 5
This should be the circumference of the Earth in meters
________________m
Step 7: Divide Step 6 by 2π (Note: 2π = 2 x 3.14 = 6.28)
This is the radius of the Earth in meters
________________m
The accepted value of the Earth’s radius is 6.37 x 106 meters. Using this value calculate
your percent error for your measure of the size of the earth. Recall:
Step 8: Find the percent error between your value for the radius of the Earth and the
accepted value of 6.37 x 106 meters.
________________%
Repeat the same set of calculation for Point C and Point D.
Step 1: Write down the latitude for Point C (Southwest Corner):
________________
Step 2: Write down the latitude for Point D (Northwest Corner):
________________
Step 3: Subtract the value of Point C from the value of Point D:
________________
Step 4: Divide 21600 by Step 3.
This is number of “pie slices” in our earth.
________________
Step 5: Write down the distance between Point C and Point D.
This value should be in meters.
_______________m
Step 6: Multiply Step 4 by Step 5
This should be the circumference of the Earth in meters
________________m
Step 7: Divide Step 6 by 2π (Note: 2π = 2 x 3.14 = 6.28)
This is the radius of the Earth in meters
________________m
The accepted value of the Earth’s radius is 6.37 x 106 meters. Using this value calculate
your percent error for your measure of the size of the earth. Recall:
Step 8: Find the percent error between your value for the radius of the Earth and the
accepted value of 6.37 x 106 meters.
________________%
Questions.
State your answer in complete sentences.
1. Calculate the Earth’s radius using Eratosthenes’s two possible values of 46,620
km and 39,690.
2. Using the acceptable mean value of 6370 km for the earth’s radius. Find the
percent error for the to values found in question one.
3. How does your percent error compare with Eratosthenes’s percent error? Why do
you think his values have a wide margin for error?
4. What causes some of the errors in your values for the size of the earth?