Name: ________________________________ Class:___________________ Date: ____________
Earth Science –Mrs. Gill-Klesaris
Earth’s Dimensions
Eratosthenes Circumference Lab
Earth Science: Calculating the Circumference of the Earth
Around 240 BCE, a Greek mathematician living in Egypt named Eratosthenes used
a clever method involving observations of the sun in two nearby locations to
determine the circumference of Earth. In this lab, you will use modern technology
to collect data that you can use to determine the size of the earth using a similar
technique to Eratosthenes. Our goal is to see if modern technology can “beat” the
accuracy of this ancient technique.
Materials and descriptions:
Open field to collect data – We are going to use the field behind the
stadium.
GPS Tour app on phone- Download this app on your IOS device via the App Store. Set your GPS to report
your position in decimal degrees (DD.DDDD) and to report distances in meters, this is the apps default, but
it can be changed using the “i” information button in the bottom right corner.
Surveying tape measure- this is a 100 foot tape measure- it includes metric units.
Group created data sheet- each group is going to be responsible for deciding what data to collect, how to
collect it, and how to report it and ultimately how to analyze it.
Part 1: Understanding how Eratosthenes made his Calculation
Diagram #1: Map of Egypt
Historic Background: Over 2260 years ago,
Eratosthenes, the Chief Librarian in Alexandria,
Egypt made a surprisingly accurate measurement of
●
the Polar Circumference of the Earth. While
visiting Syene, a city due south of Alexandria,
located on the Tropic of Cancer. There he noticed
that the sun reflected in the bottom of a deep well
at solar noon. In other words there was no shadow!
Perplexed, Eratosthenes returned to Alexandria on
the Northern Coast of Egypt to further investigate
the phenomenon.
Syene
1) Think about it, where must the sun be located in
●
the sky in order for it not to cast a shadow and
reflect in the bottom of a deep well?
_______________________________________
_______________________________________
2) Use the scale on Diagram#1: Map of Egypt to determine the distance between Alexandria and Syene:
____________________________Kilometers _____________________________ Miles
As depicted in diagram #2, the sun is so far away that the rays hit the earth parallel to each other and thereby
create different angles with the Earth’s curved surface. On Diagram 2, which city is receiving the “Direct Ray”
where the sun makes a right angle with the earth’s surface? ____________________________________
Diagram #2:
Diagram #3:
‘
The following year in Alexandria, he used the shadow casts by an Obelisk at noon on June
21st to calculate the angle between the Sun’s ray and this obelisk (angle α). Using the laws
of
trigonometry (law of similar triangles) he figured out that this angle α is equal to the
interior angle between Syene and Alexandra (angle α’). See Diagram #3.
Using the angle between the two cities and the distance between them, he could easily calculate the circumference
of the Earth, using simple geometry. Eratosthenes calculated the angle the sun’s ray’s make with the obelisk in
Alexandria to be 6.0 degrees (about 1/50th of a circle).
3) Using this data and the distance between the two cities what is Eratosthenes polar circumference of Earth in
kilometers? Show work here:
Hint: Since there are 360 degrees in a circle, we can set up a proportion to use the distance between Alexandra
and Syene and the angle created with the sun to calculate the circumference.
𝐴𝑛𝑔𝑢𝑙𝑎𝑟 𝐷𝑒𝑔𝑟𝑒𝑒 𝑏𝑒𝑡𝑤𝑒𝑒𝑛 𝑐𝑖𝑡𝑖𝑒𝑠
360˚
=
𝑑𝑖𝑠𝑡𝑎𝑛𝑐𝑒 𝑏𝑒𝑡𝑤𝑒𝑒𝑛 𝑐𝑖𝑡𝑖𝑒𝑠
𝑐𝑖𝑟𝑐𝑢𝑚𝑓𝑒𝑟𝑒𝑛𝑐𝑒
Part 2: Determining the Polar Circumference of Earth Using Modern Technology.
Procedure:
1. Brainstorm with your group a procedure for using the equipment provided to measure the circumference.
2. On a single sheet of paper develop step by step instructions for collecting the data. Your procedure should be
clear enough that it can be reproducible and followed by another group.
3. Decide what data needs to be recorded. Draw a data table in the box A below to organize the data you plan to
collect.
4. Derive a formula for calculating Polar Circumference using your data. Show all your work below including the
formula in box B below.
Box A:
Box B:
Discussion Questions:
Find the accepted value by performing a web search for “polar circumference of the Earth”
Use the data in this lab to complete the chart below (make sure to report results in kilometers):
Eratosthenes calculated Polar Circumference
Student Calculated (or observed) Polar circumference
Earth’s actual Polar Circumference
How many kilometers were you off from the actual Circumference? ___________________________
How many kilometers was Eratosthenes off from the actual Circumference? _____________________
Calculate the percent of error in your calculation of the Earth’s circumference (show your work and the formula,
even if you use a calculator).
Calculate the percent of error in Eratosthenes calculation of the Earth’s circumference (show your work, even if
you use a calculator).
Part 3: How does the Equatorial circumference of Earth compare to Polar Circumference?
Let’s take a boat trip on March 21st to the equator just off the West coast of Africa, where it intersects the
prime meridian. Since our boat is on the prime meridian the sun will be at the zenith high noon occurs at 12:00 noon
Greenwich Mean Time (GMT). Next we travel 1,669.83 km due West and record that high noon (sun at the zenith)
occurs 1 hour later at 1:00 PM (GMT).
HINT: If a circle has 360°, and a day has 24 hours; how many degrees does the earth spin in an hour? ___
Hmmm, so if we know the degrees covered in hour, and we know the distance covered along the equator in an
hour, can we set up a proportion to figure out the distance covered in the 360 degrees around the equator?
Show Work Here:
Based on this information, what is the Equatorial Circumference of Earth?_________________________
Degree of Flattening= =
𝑑𝑖𝑓𝑓𝑒𝑟𝑒𝑛𝑐𝑒 𝑏𝑒𝑡𝑤𝑒𝑒𝑛 𝑝𝑜𝑙𝑎𝑟 𝑎𝑛𝑑 𝑒𝑞𝑢𝑎𝑡𝑜𝑟𝑖𝑎𝑙 𝑐𝑖𝑟𝑐𝑢𝑚𝑓𝑒𝑟𝑒𝑛𝑐𝑒𝑠
𝑒𝑞𝑢𝑎𝑡𝑜𝑟𝑖𝑎𝑙 𝑟𝑎𝑑𝑖𝑢𝑠 𝑜𝑛 𝑝𝑎𝑔𝑒 15 𝑖𝑛 𝐸𝑆𝑅𝑇
𝑥100%
Use the equation above to determine the degree of flattening?__________________
Is this a noticeable difference when looking at a satellite image of earth from space? Yes or No?
Sample/ Suggested Procedure
1. Locate yourself in an open area, say near the center of one of the playing fields, and set a waypoint by
recording your latitude and longitude data in decimal form using the GPS Tour App.
________________________________________________________________
2. Walk in such a way that your latitude changes, but your longitude remains constant. In what direction(s) can
you walk to accomplish this? ______________________
3. Layout the tape measure as close to a North-South direction as possible, for a distance between 50 meters
and 100 meters. Record Distance: _________________________
4. Check that your new location has exactly the same longitude as your original location, and set a second
waypoint. ________________________________________________________
5. Determine the angular distance between the two points by finding the difference between the latitudes of
the two waypoints. Record the angular distance: ________________________
6. Use constructed formula to calculate the polar circumference.
Sample GPS FIELD DATA
1. First observation point given in degrees, minutes, seconds: _________________________
First Observation Point in decimal degrees:___________________________________
2. Second observation point given in degrees, minutes, seconds: _______________________
Second observation point in decimal degrees:___________________________________
3. Angular Measurement (difference in latitude) between points = ____________________
4. Linear distance between Observation Points: ____________m
= ___________ km
5. Calculate the Circumference of the Earth:
a. set-up the formula:
b. solve (circle your final answer):
The actual circumference of the Earth is 40,008 km. Were you close?