Eratosthenes:
Estimating the Circumference of the Earth.
Subject: Mathematics
Topic: Geometry
Grade Level: 8-12
Time: 40-60 min
Pre Show Math Activity
Introduction:
In Show Math, students will learn how in 200 BC, the Greek mathematician Eratosthenes calculated the
circumference of the Earth using simple geometry and a few measurements. His estimate of the circumference
was remarkably accurate—to within 2% of the real value. In this activity, students will follow in Eratosthenes’
footsteps by finding the circumference of the Earth themselves using sticks, shadows, and abstraction. It is an
activity of discovery, where the students will piece together the geometry for themselves, using Eratosthenes’s
assumptions and the few pieces of information he had available.
Learning Objectives:
By the end of this activity students should:
• Understand how abstraction can help solve math and real life problems
• Explore the problems encountered when measuring the Earth’s circumference
• Examine the ways that errors will occur by measuring circumference in this way
Skills:
Students should develop:
• Team work skills
• Communication adn collaboration skills
• Abstract thinking skills
Material & Resources:
• Chart Paper
WNCP Curriculum Links:
Mathematics 9
Solve problems and justify the solution strategy using circle
properties. [C, CN, PS, R, T, V]
Apprenticeship and Workplace
Mathematics 10
Analyze puzzles and games that involve spatial reasoning, using
problem-solving strategies. [C, CN, PS, R]
Solve problems that involve parallel, perpendicular and
transversal lines, and pairs of angles formed between them.
[C, CN, PS, V]
Foundations of Mathematics 10
Develop and apply the primary trigonometric ratios (sine, cosine,
tangent) to solve problems that involve right triangles.
[C, CN, PS, R, T, V]
Foundations of Mathematics 11
Solve problems that involve the properties of angles and
triangles. [CN, PS, V]
Apprenticeship and Workplace
Mathematics 12
Demonstrate an understanding of the limitations of measuring
instruments and solve problems. [C, PS, R, T, V]
Background:
In ancient times it was widely believed that the world was flat with the
heavens a physical dome spanning over it. However, there were some
early arguments for a spherical Earth. For example, it was observed that
during a lunar eclipse the Earth cast a circular shadow of the Earth on the
moon during a lunar eclipse. It was also noted that the star Polaris is seen
lower in the sky as one travels South.
Pythagoras observed that ships masts appear before their hull when they
appear over the horizon. Since these phenomena are observed everywhere,
it suggests that the earth is curved everywhere. The logical result is that the
earth must be spherical.
Archimedes and Plato both believing the Earth
to be a sphere, tried to estimate the circumference of the Earth. However, the first to give an
accurate calculation of the circumference of the
Earth was Eratosthenes in about 240 BC. Eratosthenes knew that in Cyene, Egypt (now known
as Aswan) on the summer solstice, when
the sun its’ zenith, sticks and pillars cast no
shadow. However, in his home town of Alexandria, he saw that a stick planted in the ground
cast a shadow measuring 7.2 degrees. Since the
sun is very far away, when its rays hit the Earth
they are nearly parallel. Therefore, if the Earth
were flat and if a stick in Cyene cast no shadow
at noon, then a stick in Alexandria would also
cast no shadow. Because the stick in Alexandria
does cast a shadow, he could use this information to estimate the earth’s circumference
Imagine we extend the sticks until they meet at
the centre of the Earth. We can see that since the
sun’s rays are parallel and that alternate angles are
equal, we can see that Alexandria and Cyene are
separated by an angle of 7.2 degrees.
Eratosthenes paid someone to pace out the distance between Cyene and Alexandria which he found to
be 5,000 stadia - approximately 800km. Now using proportion he found the circumference using the
following equation:
Eratosthenes did some rounding to 700 stadia per degree and implied a circumference of 252,000 stadia. If Eratosthenes used the Egyptian stade of about 157.5 metres his measurement turns out to be 39,690 km, an error of
less than 2%. The circumference of the Earth between the North and South Pole is 40,008 km.
There are limitations in the accuracy of Eratosthenes’ calculation. The accuracy of his measurement is
reduced by the fact that Cyene is not directly south of Alexandria and that the sun rays are not perfectly
parallel when they hit the Earth. However, the greatest limitation of Eratosthenes’ method is that his
distance measurement was not reliable because the roads connecting the two cities were not a straight
line path (great arc, actually) between them. Given the margin of error in these aspects of his calculation, the
accuracy of Eratosthenes’ circumference estimate is surprising.
Resources:
• https://en.wikipedia.org/wiki/Eratosthene
A description of Eratosthenes’ experiment
• http://en.wikipedia.org/wiki/Flat_earth
Details the history of the beliefs about the Earth’s shape and size
• http://www.youtube.com/watch?v=0JHEqBLG650&feature=player_embedded
A video that explains Eratosthenes’ methods
Activity Instructions:
1. To class say: “Today we are going to go back in time to Ancient Egypt. We need to find the circumference
of the Earth without any modern technology. Any ideas how we could do this?”
2. Tell class that in Ancient Times, Eratosthenes used a few observations and geometricalreasoning to deter
mine the circumference. “Put yourself in the Eratosthenes’ shoes. Here’s what you know and what you
assume:” Display the overhead (Figure 1) of information available to Eratosthenes. You may also want to
print off some copies of Figure 1 for the groups.
3. Divide class in groups of 4-5 students. Given them each a piece of chart paper and have each group draw a
labelled diagram of the sticks in the earth and the shadow.
4. Allow each group to share their diagrams with the class. If necessary, provide hints using the overhead of
the geometry (Figure 2), in case the student diagrams aren’t quite right.
5. Tell the students to go back into their groups and tell them to use the
diagram to find the circumference of the Earth. Circulate through
out the class and help groups as they need it.
6. Allow each group to share their calculations. If necessary solve the
problem on the board.
7. Explain to the class that there weren’t actually two sticks but that
Eratosthenes simply noticed that no structures in Cyene—such as
pillars—cast a shadow at solar noon (which is when the
sun is at its highest point in the sky for a particular day) on the
Summer Solstice (on which day the noontime sun is higher in the
sky than any other day of the year). He carefully measured the
length of a shadow in Alexandria for an object of known height at
noon on the same day.
8. Hand out take home worksheet.
Discussion:
To aide you in your discussion you may want to use the overhead
(Figure 3) of the map, which will help illustrate the error in pacing out
the distance between Cyene and Alexandria and that Alexandria and
Cyene are not on the same meridian. Today we know the circumference
of the Earth is 40,008 km, where does the error come from? Engage in
class discussion about the errors that may occur in Eratosthenes method.
Discuss ways we could find the circumference in this day and age.
What was the hardest step in finding using this information to find the
circumference of the Earth?
Ancient Measurements
You may notice that the measurement of the stick and the shadow
are not in centimetres but in cubits.
This was the measurement used in
Eratosthenes. Although, it is not
necessary to convert from cubits to
centimetres to solve the problem,
students may want to know. A cubit
is 45.72 centimetres.
Alternate Angles are Equal
You may want to remind the class
that Eratosthenes knew that given
two parallel lines, the alternate
interior angles are equal.
If the class is having trouble, ask
them where there are two parallel
lines in their diagrams?
Possible sources of error Include:
• The measurement of the distance between Cyene and Alexandria was
inaccurate - someone paced the distance and there was no straight path
• Cyene and Alexandria are not on the same meridian - the geometry of
Figure 2 assumes the cities are north/south of each other. If the cities are
on different longitudes, the distance measurement between overesti
mates, giving a larger estimate of the circumference. Imagine, for
example, that instead of Alexandria, Eratosthenes had used another city
far away in West Africa, but on the same latitude as Alexandria. At noon,
the shadow’s angle would still be 7 degrees, but the distance between the
cities would be several thousand kilometres, providing an enormous
estimate of the circumference. A good way to illustrate this to the class
would be to use a globe and lamp.
• Measuring the length of a shadow precisely can be difficult - hard to see
a distinct edge.
• The sun’s rays are not perfectly parallel - the sun is a fixed point very far
away but not infinitely far away.
Measuring Shadows with
Your Class
There are online projects where you
repeat Eratosthenes experiment
with your class by measuring
shadows and partnering with
another school in a different part
of the world. The process is a bit
more involved and will take a
couple class periods. To find out
more visit: http://www.ciese.org/
curriculum/noonday/index.html
Figure 1 - Information for Finding Circumference of the Earth
Eratosthenes assumed :
• The earth is a sphere.
Therefore a stick planted
vertically in the ground will
be perpendicular to the earth.
• The sun is very far away. When its rays hit the Earth
they are parallel.
• Alexandria is directly north of Cyene—they lie on the
same meridian.
Eratosthenes knew:
• At noon in Cyene, the sun is directly overhead and casts
no shadow.
• The distance from Cyene to Alexandria is approximately
800km.
• The stick used in Alexandria measures 4 cubits long and its
shadow at noon is 0.505 cubits.
Figure 2 - Eratosthenes’ Diagram
Figure 3 - The Distance Between Cyene and Alexandria