Measuring the Earth
Grade Level: 9-12

Overview Many students have little understanding or appreciation for the process of science. Students should be involved in doing science, taking measurements, collecting data at the very beginning of the school year. This activity presents an opportunity for community participation.

Purpose The purpose of this activity is to get students interested and involved in doing science; give them a reason to use some of the math they have learned; and develop a feeling of cooperation in working with people from other schools.

Objectives Students will be able to:
i. Work effectively in a small group to take accurate measurements at a specific time.
ii. Apply their knowledge of geometry and trig. to determine the measure of an angle.
iii. Use significant digits in their reports.
iv. Calculate percent error.
v. Use their research skills to determine accepted values.

vi. Demonstrate the value of cooperation in achieving a common goal.

Resources/Materials
A meter stick or measuring tape

Scientific calculator.

Activities and Procedures The following is an important Background Information for the teacher. This activity goes beyond what children will ordinarily encounter in their schoolwork. It is demanding in terms of both the activity and the calculations.

Eratosthenes, a Greek mathematician, was the first to measure the circumference of the earth. He based his measurement of the earth on the assumptions that the earth was round and the sun's rays are parallel. He knew that at noon on the day of the summer solstice in Alexandria, Egypt, a vertical post cats a shadow. At the same time in Syene, a town directly to the south, a vertical post casts no shadow. Eratosthenes used Euclidean geometry to determine that the angle formed by the post and an imaginary line from the end of the shadow to the top of the post equals an angle at the earth's center formed by imaginary lines from the two towns. He calculated the earth's circumference by measuring the distance between Alexandria and Syene, and multiplying it by the number of times the angle at the earth's center is contained in 360 degrees.
i. Contact a class directly north or south of you (in a different state if possible) and set a specific date and time to take the measurements.
ii. Divide the class into groups and practice at least once before the day of the activity. They are to measure the height of an object (a pole is good) and the length of its shadow at a specific time. This activity should be started 15 minutes before the stated time.
iii. Assign several students to research the circumference of the earth and others to find several ways to determine the distance from your school to the other group's school (maps, auto clubs, etc.). Eratosthenes had a slave to pace off the distance between the two cities and report back to him.
iv. The measure of the angle is found by dividing the length of the shadow by the height of the object on your scientific calculator and then push 2nd function tangent. However, this is not the central angle. The angle from the other school must be subtracted from your angle and the absolute value of this difference is the central angle. The circumference of the earth can them be calculated by setting up a ratio and solving for the circumference. The following formula can be used:
 
central angle
360 degrees
=
distance from schools
circumference

v. The students will have to decide how many significant digits to use in their results and then calculate the percent error from the value they found in their research.

Tying it all together
i. Discuss the sources of error and the fact that your results depend on other students making accurate measurements.
ii. If available, you should show the first tape of the "Cosmos" which tells about Eratosthenes.

3. The next activity might be to indirectly measure the height of a flagpole.

Assessment This is an activity that involves a number of students. Try to find out what aspect of the activity solicits active group participation.

Suggestions/Modifications

  • Teacher may use creative ideas of distances to measure around the school, village, or local areas.
  • Calculations can be done with one calculator or can be estimated.
  • The students may need to be motivated to complete the assignment through a contest or a game.
Author(s) Jane Rich, Shawnee High School, Shawnee, OK