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Name                                                                                                  Date

Mrs. Luna                                                                                            Period

Lab # 5                                                                                              Date Due _______

# of Minutes: ______                                                                     Lab Group: _____

 

 

 

                                                LAB # _____________

 

                                    ERATHOSTHENES AND EARTH’S CIRCUMFERENCE

 

 

Introduction:  More than 2000 years ago, a man named Eratosthenes made a surprisingly accurate estimation of Earth’s circumference.  He did this by using some simple geometric relationships.  In this lab, you will use Eratosthenes’ method to determine the circumference of a circle.

 

 

Materials: 

·         Blank sheet of paper

·         Safety compass

·         Protractor

·         Flexible Metric Ruler

·         Pen

·         Pencil

·         Calculator

 

Define the following word: 

 

Model:______________________________________________________________________________________________________________________________________________________________________

Circumference: ____________________________________________________________________________________________________________________________________________________________________________

 

Procedure: 

 

Part A:  Finding a Circle’s Circumference

 

  1. Visually locate a point as close to the center of a blank sheet of paper as possible.  Mark the point and label it Point C.  With a safety compass, draw a large round circle around point C.

 

  1. Use the ruler to draw a straight line in any direction from point C to the edge of the paper.  Label the point where the straight line intersects the circle point A.  The line connects points A and C is called line AC.

 

  1. Place a protractor along line AC so that its center is on point C.  Mark off an angle between 150 and 500.

 

  1. . Record your angle in Table 1.

 

  1. Complete the angle by drawing a straight line from point C through the point you have marked off with the protractor to the edge of the paper.  Label the point where this straight line intersects the circle point B.  The line connecting points B and C is called line CB.

 

  1. Set the flexible metric ruler on edge and bend it to follow the circumference of the circle.  Use the curved ruler to measure, to the nearest tenth of a centimeter, the length of the circle from point A to Point B (an arc).  Record your value in Table 1.

 

  1. Measure the length from point C to point A, to the nearest tenth of a centimeter.  Record the value in Table 1.

 

  1. Answer Questions 1 and 2 under Analysis and Conclusion.

 

Part B:  Finding the Earth’s Circumference

 

Eratosthenes used careful observations of the sun’s rays to find Earth’s circumference.  The figure on the next page shows the sun’s rays striking two locations at Earth’s surface, E and F.  At point E, the sun is straight overhead and strikes the strikes the surface at a 900 angle.  At point F, the sun’s rays strike the surface at angle GFH.  Measure angle GFH and record in a copy of Table 2.

 

  1. Because the sun is far away, Eratosthenes considered its rays to be parallel.  Bases on geometric principals, the parallelism means that angle IFH equals the angle between E and F at Earth’s center.  To find IFH, subtract GFH from IFH.  Note that angle IFG is a right angle (900 ).
    1. Using the curved ruler technique from Part A, measure the distance from point E to F along the segment of Earth’s surface shown to the nearest tenth of a centimeter.  Record the arc length in Table 2.

     

    1. Use the scale 1 cm =1,800 kilometers to convert the length of arc EF to kilometers. Record this value in Table 2.

     

    1. Answer Questions 3 and 4 in Analysis and Conclusion.

     

     

                                                                               Table 1

    Angle used ( ) =

     

    Length of arc AB (cm) =

    ( distance along circle)

    Length of line AC (cm) =

    (radius of circle)

     

     

     

     

     

     

                                                                           Table 2

     

    Angle GFH ( ) =

     

    Angle IFH () =

     

    Measured length , arc EF

    (distance along circle in cm)

     

    Distance, E to F (km)

     

     

    Analysis and Conclusion

     

    Part A: Finding a Circle’s Circumference

    Show all work

     

     

    1.       The length of arc AB has the same relationship to the circumference of the circle as the angle used has to the whole circle (3600 ).  Use the equation below to find the circumference of the circle.

     

     

    Arc AB    =  angle used

                         Circumference     3600

     

     

     

     

     

     

     

    2.      Using the standard formula C = 2πr, determine the circumference of your circle again.

     

     

     

     

     

     

     

    3.      Calculate the circumference using the following equation and the information provided in Table 2

     

     

    Distance EF    =   angle IFH

        Circumference        3600

     

     

     

     

     

     

     

     

     

     

     

     

    4.      Earths’ actual circumference is approximately 40,000km.  Determine the percentage by which your answer differs from Earth’s actual circumference using the equation below.

     

     % = actual circ.  – ans.    X  100

                      Actual Circ.

      

 

           

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