# Duluth, Georgia USA

**March 2007**

Flag | City | Country | School | Latitude (°) +N/-S | Longitude (°) +E/-W |
---|---|---|---|---|---|

Duluth | Georgia USA | Charles Brant Chesney Elementary | 33.983 | -84.150 |

Date | Gnomon | Shadow | Angle |
---|---|---|---|

March 21 | 100 cm | 59 cm | 30°32’ = 30.541° |

On March 21, 2007 the Spring Equinox, at 1:44 p.m. you could find Mrs. Fox’s 5th grade gifted math class outside attempting to duplicate an experiment that a Greek mathematician and scientist named Eratosthenes conducted over 2,000 years ago.

We were measuring the Sun’s angle as a part of a greater experiment of measuring the circumference of Earth.

The tools that were used in measuring the Sun’s angle were a meter stick, which acted as a transversal through the parallel lines that were the sun’s rays. In Geometry we studied Euclid’s Parallel Postulate. Eratosthenes used this concept and used it to measure the Sun’s angle.

One tape measure 150 cm long was used to measure the length of the shadow cast by the meter stick. The other tape measure was taped to the top of the meter stick and formed a ’hypotenuse’, creating an acute angle that increased and decreased as the position of the Sun changes. Each team used a long piece of butcher paper to better see the shadow cast by the meter stick. We used writing utensils to record our shadow and Sun’s angle measurements. We later used a table, created on our laptops, to record our measurements. We used a protractor to measure the angles.

One team created a base for holding the meter stick at true vertical using a glass vase filled with rocks. The problem was that because the circumference of the base of the vase was wide, we couldn’t place the device for measuring the shadow length right at the bottom of the vertical meter stick. The stand was in the way! One team tried using a plastic water bottle filled with crumpled paper. The problem with that was that the bottle and paper were too light weight for the height and weight of the wooden meter stick. It just kept falling over even when we
tried to tape it to the sidewalk. We also could not get an accurate measurement of the shadow length. Another team tried to anchor their
meter stick with Playdoh, but could not maintain true vertical even after they tried supporting the Playdoh with rocks. We all ended up just holding our sticks upright.

We began our first measurement at 1:31 p.m., about 13 minutes before true noon. Our first shadow length measurements ranged from 59 cm to 63 cm. Our first Sun’s Angle measurements ranged from 28 degrees to 32 degrees. Our shadow length at True Noon ranged from 64 cm to 72 cm for an average of 68 cm and the Sun’s Angle ranged from 32 degrees to 38 degrees. We threw out the 38 degrees as an outlier and found an average of 32.5 degrees.

We subtracted our Sun’s angle from Liberty’s angle and got 4.832 degrees which we then multiplied by 111, the distance in kilometers for one degree of Latitude on Earth. Our product was 536.352 kilometers. Next, we multiplied 536.352 (the distance in kilometers between our two cities) by 360 (the number of degrees in the Circumference of Earth) and the result was 193,086.720, then divided this result by the difference in our Sun’s angles (4.832) and got the Circumference of Earth as 39,960 km! The true Earth’s circumference is 40,075.16 km. The percentage of error was .03%, less than one percent! Our partners at Logos Christian Academy are located at 80.38W Longitude while we are at 84.01 which is very close in degrees longitude.

We also chose Conners-Emerson’s 5th Grade gifted reading class in Bar Harbor, Maine and using their data subtracted our
Sun’s angle from theirs with a result of 9.75 degrees. Next we subtracted Duluth’s Latitude (34.01) from Bar Harbor’s (44.39) which is
10.38 degrees. The formula we use looks like this ratio:
Difference in Sun’s angles / 360° = Distance in Latitude/C(circumference)

We then took 10.38 degrees, multiplied it by 111 km which equals 1152.18 km and then cross multiplied this distance on Earth, measured in degrees latitude, by 360 to get 414,784.8. All that was left to do to complete the ratio was to take the difference in the Sun’s angles and divide it by 9.75 which gave us Earth’s Circumference at 42,542.03 km, an error of 6.15%. We think the reason is because of our difference in Longitude. We are at 84 W. Bar Harbor is at 68.21W.

These are the comments of team members who participated in this experiment:

“This was a magnificent experiment that taught us how you should be accurate with measurements and calculations, otherwise if you get something wrong, it can ruin someone else’s results on not only this experiment, but if they use your data for another purpose!”

“It was an interesting experiment that helped us to perfect our skills in measurement, math calculations and science, and it helped us to improve our teamwork.”

“This experience reminded us about the relationship between math, science, and technology.”

“The experiment taught me how to measure the Sun’s Angle.”

“We all agree that we could conduct this experiment anytime, anywhere on Earth as long as the sun is shining.”

Use this link to look at the 2005 Final Report and pictures, 2007 pictures soon to be posted AND Mrs. Fox’s Webquest "Earth, Eratosthenes, and the Equinox"! http://www.chesneyelementary.org/gi... We feel as if we are true scientists and true mathematicians like Euclid and Eratosthenes.

Chesney Elementary 5th Grade FOCUS 2007

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