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Lesson: Measure the Earth’s Circumference


(This lesson combines material from Youth Net and the Center for Innovation in Engineering and Science Education (CIESE) and Noonday Sun project. The latter is made available with permission of the Stevens Institute of Technology; copyright 2011 the Trustees of the Stevens Institute of Technology, Hoboken, NJ 07030).

Level: Grades 5-12. Time Required: One to two class periods.

Overview

In this project, students in grades 5-12 will recreate the remarkable measurement of the circumference of the earth first performed over 2,000 years ago. Using rulers, protractors, and meter sticks, students measure shadows cast by a stick on a day close to the equinox, then use that measurement in a simple equation to determine the circumference.

In addition, the class will join students from around the world who will also make measure the Earth’s circumference then submit their results to a joint project Website.

Materials

  • a meter stick or vertical shaft
  • a large piece of paper
  • material to help hold the stick steady (a metal book end, a liter plastic bottle anchored by send, a large ball of clay, etc.)
  • a carpenter’s level (optional)
  • rock and string (optional)
  • a compass
  • a local newspaper or the Naval Oceanography Portal to check sunrise/sunset times
  • a globe or atlas to determine the distance between your location and the equator

Background

By the third century BC, scholars believed that the earth was a sphere. One empirical piece of evidence came from observing ships as they approach the horizon, since the sails appeared to dip into the ocean. But no one knew for sure how big this sphere was. It took a curious and ingenious librarian named Eratosthenes to discover a remarkably simple method for measuring the circumference of the earth.

EratosthenesOver 2,000 years ago the Greek geographer Eratosthenes (circa 276 to 194 B.C.), made a remarkably accurate measurement of the earth’s circumference. In the great library in Alexandria, he read that a deep vertical well near Syene, in southern Egypt, was entirely lit up by the sun at noon once a year. Eratosthenes reasoned that at this time, the sun must be directly overhead, with its rays shining directly into the well. In Alexandria, almost due north of Syene, he knew that the sun was not directly overhead at noon on the same day because a vertical object cast a shadow. Eratosthenes could now measure the circumference of the earth (sorry, Columbus) by making two assumptions — that the earth is round and that the sun’s rays are essentially parallel.

500px-EratosthenesHe set up a vertical post at Alexandria and measured the angle of its shadow when the well at Syene was completely sunlit. Eratosthenes knew from geometry that the size of the measured angle equaled the size of the angle at the earth’s center between Syene and Alexandria. Knowing also that the arc of an angle this size was 1/50 of a circle, and that the distance between Syene and Alexandria was 5000 stadia, he multiplied 5000 by 50 to find the earth’s circumference. His result, 250,000 stadia (about 46,250 km), is quite close to modern measurements. (from Investigating the Earth, AGI, l970, Chapter 3, p. 66.)

Eratosthenes’s formula:

D                    A
_____ = _____

d                     a

d = distance between Syene and Alexandria
A = 360 degrees assumption of round earth
a = shadow angle of vertical stick
D = to be determined (circumference)

Procedure

Tips:

  • Perform the measurement as close to the equinox as possible, though any time within a two-week span will be fine, as the margin of error is very small if done within this time period.
  • Check local weather forecast to select a day that will most likely have full sunlight.
  • Perform the measurement when the sun is highest in the sky. See below for determining your local solar noon time.
  • Caution students that to measure the shadows accurately, they must keep the meter stick firmly vertical. Wind can be a major deterrent.
  • Use paper large enough to note where the end of the shadow falls. A compass comes in handy to determine the direction in which the shadow falls. Since the edge of the shadow is “fuzzy” and the shadow will move from west to east (if in the northern hemisphere), students should take care in deciding where to place their mark.

Before the experiment:

1. To conduct this experiment online, combining data with that from another school, teachers should register their classes with the Eratosthene Experiment or The Noon Day Project, Stevens Institute of Technology.

2. Make arrangements ahead of time so that the class can meet around the time of local solar noon.

3. Determine an appropriate outdoor location where students can conduct the activity at your school.

With the Students:

Day 1: Before the experiment

1. Introduce students to the activity by telling them about Eratosthenes, his measurement, and the equinox. Explain that the class will perform an experiment on a date close or on the equinox to measure the Earth’s circumference based on Eratosthene’s model. In addition, the class will join students from around the world who will also take measurements and submit their results to a joint project Website.

2. Divide the class into 4 or 5 working groups. Assign each group to measure the distance between your location and the equator, then reconvene to determine the correct answer. This will serve as the latitude measurement for your site.

3. Check a local newspaper or the online Naval Oceanography Portal to determine sunrise/ sunset times for the day of the experiment. Have each group  calculate the midpoint of these two times to determine the exact time of the solar noon for your location and time zone. Compare results to determine the correct answer.

Day 2: Conducting the Experiment

1. Gather the students and materials about 20 minutes before solar noon, then take the class to the spot where the experiment will be conducted.

2. Set up the measuring station(s) — teachers can conduct a single experiment or have each group set up its own measuring station. Give students the following directions:

a) Set up the measuring stick, ensuring that it is perfectly vertical. It may be taped to a metal book end, set in sand inside a liter plastic bottle, set in a large ball of clay, etc.

b) Place a large piece of paper flat on the ground under the measuring station to mark where the shadow ends. Since the edge of the shadow is “fuzzy” and the shadow is moving from west to east (if conducted in the northern hemisphere),  be careful in deciding where to place your mark. (Student may find it interesting that the shadow points towards the north. But does it point to true or magnetic north? A compass will come in handy to determine this.)

b) Hold the meter stick perfectly vertical.

c) Mark on the paper the end of the shadow at one-minute intervals over a ten- to twenty- minute period beginning at least 10 minutes before solar noon. Remind students to measure and record the length of the shadow cast by the meter stick to the nearest centimeter.

d) Using statistical computations, the class should arrive at what they feel is the length of the shadow cast at local noon, the time of the shortest shadow.

Inputting the Experiment Results

1. After concluding their measurements, students should gather their materials and return to the classroom. The groups should share their results, with the teacher recording the values on the board. Assuming there will be different values, students need to determine their “best” shadow length and decide which will represent the class’s best estimate of the shadow length at local noon time.

2. The length of the shadow at local high noon, along with the latitude for your site will be input to a joint project Website, where the class can find similar information from another school, then use a simple proportion to make a fairly accurate calculation of the Earth’s circumference determined by the pair of sites.

3. With the teacher’s assistance, the class should complete its report, submitting its data to either The Eratosthene Experiment Report Page or The Noon Day Project, Stevens Institute of Technology. Note: Teachers should fill out the online registration of one of these sites several days before the experiment.

 

Day 3: Analyzing Data, Determining Circumference and Submitting Data

1. To complete the calculation of the Earth’s circumference, the class will use the data it collected in combination with data from another school…. Follow the procedure described below.

a) Determine the length of the shadow at each location at the point of local noon.

b) Create a precise scale model of the triangle configuration of each location, based on the shadow measurements, then determine the sun angle for each using a protractor. Calculate the central angle by estimating the difference between these angles.

c) Submit your data online using steps found here.

Example Circumference Calculation

If you can measure the sun’s angles at two different positions on the earth at the same time, you can figure out the central angle (Angle A, B, C below). Then you need the distance along the surface from A to C. This gives you the dimension of one slice of the cross section of the Earth. How big is the whole circumference?

Here is an example to demonstrate how this information leads to determining an empirical value for the circumference of the earth. The two sites will be at Manasquan, New Jersey, and San Juan, Puerto Rico. At local noon on the same day, the experimenters measured the shadow cast by a meter stick. In Manasquan, the shadow length equaled 80.5 cm. In San Juan the shadow length was 35.3.

The next step is to figure out the sun angles at Manasquan and Puerto Rico. Create a scale model of your triangles and then directly measure the angle. In this case, we would create a triangle for Puerto Rico with sides that are 3.53 cm and 10.0 cm (or 3.53 inches and 10.0 inches). As long as the proportions are the same the angle should be the same. A protractor measures the angle.

The central angle equals the difference between these angles. Using these measurements, the central angle is 38.85 – 18.59 or 20.26 degrees.

Now that we know the central angle, we can determine how many such angles would make up the full circle. That is, how many “slices” can we make where the angle is 20.26 degrees? Since the total is 360 degrees, then you would have a little less than 18 equal “slices” (or 17.77 to be more precise.) Next, each slice’s edge represents the distance between Manasquan and Puerto Rico, or more accurately, the North-South distance between the two sites.

Manasquan, New Jersey: Location: 40:07:34 (40.13) N 74:02:59 (74.05) W

San Juan, Puerto: Location: 18:28:00 (18.47) N 66:07:00 (66.12) W

The distance between these two places along a north-south line is approximately 21.7 degrees or 2404 km (each 1 degree of latitude is about 111 km apart). So if the length of the “slice” is about 2400 km and there are about 18 slices then the projected circumference is 2400 x 18 = 43,200 km which is about an 8% error since the average circumference is 40,008 kms. (more precisely, it is 2404 km times 17.7 “slices”).

 

Extension

Eratosthenes Experiment Bonus Questions

Here are a series of thought-provoking ideas contributed for Youth Net by James D. Meinke, Professor of Educational Technology, Baldwin-Wallace College, Berea, Ohio.  Challenge students to answer one or more. Classes can send their answers (remember to include the question you are answering and your school information) to heratos@youth.net. If you have an idea for a bonus question, please send it to jmeinke@en.com.

Logic of the Assumptions:

It’s curious to note that Eratosthenes had other reasons for believing that the Earth was round and that the Sun’s rays were “essentially” parallel. As a bizarre twist, he could have reversed these assumptions (“The Earth is flat, and the Sun is essentially a single point with its rays being radians”) and estimated the distance between the Earth and Sun.

  1. What’s wrong (or right) about these reversed assumptions?
  2. Would the estimate of the distance be “accurate?” Why or why not?

Accuracy & Precision:

The Sun’s rays aren’t “perfectly” parallel. And the Earth isn’t “perfectly” round. And Eratosthenes & Co., didn’t have perfect synchronization (timing) for their experiment.

  1. Does “essentially parallel” mean “exactly parallel” or just “parallel enough?”
  2. What would happen if the Sun were really a single point? What if it were really much closer to the Earth? Would these changes affect Eratosthenes’ assumptions? Why or why not? Can you redesign the experiment to adjust for the changed assumptions?
  3. What are the sources of error in the techniques used in the measurements?
  4. What changes might make the readings more “exact?”
  5. Does timing of the experiments really matter?

Other Eratosthenes Calculations

Once Eratosthenes determined the circumference of the earth, he devised a method for determining

  1. The diameter of the moon.
  2. The distance between the earth and the moon.

How did he do it?


References

Relevant Websites

Other Eratosthenes Measurement Projects

[youtube]http://www.youtube.com/watch?v=0JHEqBLG650&feature=player_embedded[/youtube]

Scientist Carl Sagan describes the background of Eratosthenes’ experiment.

Image Credits: Eratosthene image and diagram, Wiki Commons. Other images made available with permission of Stevens Institute of Technology; (c) 2100 the Trustees of teh Stevens Institute of Technology, Hoboken, NJ 07030

One Response to “Lesson: Measure the Earth’s Circumference”

  1. Thanks for posting this. In my opinion this is the best math collaborative project out there. I am a bit biased. I was one of the developers of the CIESE version of this project. 🙂
    Noon Day: Measuring the Circumference of the Earth Project

    -Ihor

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