## Chapter 4 LAB Activity: Eratosthenes and the Earth’s Circumference Answers

Introduction

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Eratosthenes, a Greek geographer (about 276 to 194 B.C.), made a surprisingly accurate estimate of the Earth’s circumference. The legend follows that Eratosthenes used a deep vertical well near Syene, in southern Egypt which was entirely lit up by the sun at noon once a year as the basis of his calculations. He reasoned that at this time the sun must be directly overhead, with its rays shining directly into the well. Using two assumptions Eratosthenes was able to measure the circumference of the earth; 1) the earth is round and 2) the sun’s rays are parallel. His results concluded that the earth was: 250,000 stadia (about 46,250 km), is quite close to modern measurements.

Purpose
To calculate the circumference of a circle by using the same methodology that Eratosthenes used to calculate the earth’s circumference; deriving the relationship between arc length, radius and angle measurement.

Materials
Pencil
String
Paper
Protractor
Compass
McDougal Littell: Earth and Space Science Textbook

Procedure
Part A: See page 84 of McDougal Littell: Earth and Space Science Textbook
Part B: See page 84-85 of McDougal Littell: Earth and Space Science Textbook

Analysis and Conclusion (Part A)

Table 1
Angle used: 30°
Length of arc AB (cm): 4.1
Length of line AC (cm): 7.2

1)
(arc AB)/circumference= (angle used )/(360°) 1/circumference= (arc AB(360°) )/(angle used)

circumference= (4.1cm(360°) )/(30°) circumference= 49.2cm

2)

C=2?r^ C=2(3.14?)(7.2cm)?^
C=45.216cm

There was a deviation of 3.984cm between the two answers ascertained for the circumference of the circle. The standardized circumference equation gave a smaller value than the (arc length x 360°)/angle ratio. The second answer was probably more accurate than the first, because it uses two absolute values (Pi and 2) as oppose to the first equation which only uses one (360°). Therefore it is more probable that the first answer is not as accurate because it has two variables which require measurement and are prone to human error.

Analysis and Conclusion (Part B)

Table 2
Angle GFH: 61°
Angle IFH: 29°
Measured Length, arc EF: 2.0cm
Distance, E to F (km): 3600km

1)
(distance EF)/circumference= (angle IFH )/(360°) 1/circumference= (angle IFH )/(360°(distance EF))

circumference= (360°(3600km) )/(29°) circumference= 44689.65km

2)

%= (| actual circumference-ans.| )/(actual circumference) x 100%= (|40,000km-44689.65km|)/40,000km x 100

% Error = 11.7%

1) You can use this method that in Part B by assuming that the sun was so far off that its rays hit the earth in parallel lines. By making this assumption, that means all the sun rays running along the surface of the earth become nearly parallel. Also the assumption that the earth is shaped like a sphere is also necessary; though it is actually an oblate spheroid. Then by merely taking two reference points (cities) running along the same meridian (line of longitude) and measuring the distance between them, a radius value can be ascertained. Then during the summer solstice, when the tilt of the Earth’s axis is most inclined toward or away from the Sun, causing the Sun’s apparent position in the sky to reach its northernmost or southernmost extreme, measurements can be taken. A reading in one reference city, the sun will be directly overhead at noon on a particular day of the year. On that day at noon, vertical objects cast no shadow, and the reflection of the sun can be seen in bodies of water (i.e well). Then by measuring the angle of the shadow in the other reference city at noon on the same day when the sun was directly overhead; the angle of the shadow would be the same as the central angle of the “wedge” of the earth between city A and B. Finally by using the measurement from each city and plugging them into the mathematic relationships in Part B, an estimation of the earth’s circumference can be gained. However the problem with these measurement values is two assumptions have to be made in order for this method to be used. First the earth is not spherical and is an oblate spheroid; its shape varies from the equator to the poles and depending on the location of measurements, there can be a discrepancy between the values. Also the assumption that the sun rays are exactly parallel, which they are not; they are slightly skewed and this can result in inaccurate answers being ascertained. However surprisingly, even with these two fundamental flaws, Eratosthenes still managed to get a very accurate measurement of the earth’s circumference in the year 240 B.C.

2)
Circumference=2?r 44689.65km=2(3.14)r

44689.65km/2(3.14) =r
r=7116.18km

D=2r
D=14232.4km

x

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