**Grade Level:** 5th - 8th; **Type: **Geography

### Objective:

To find out the circumference of the earth using the shadow of the sun.

The purpose of this experiment is to see if the distance around the earth can be measured using shadows.

- Who was Eratosthenes?
- How did he discover the size of the Earth?
- How big did he think the Earth was?
- Did his contemporaries believe his theory?
- What other early models of the Earth also assumed that the Earth was a sphere?
- What early models of the Earth assumed the Earth was flat?
- Which was the more popular belief 2000 years ago? That the Earth was flat or round?
- What is true noon?

It is a common misconception that Columbus was the first person to prove that the earth was round. While many of our ancestors may have believed the Earth was flat, the truth is that scientists from many ancient civilizations had presumed that the Earth was a sphere long before Columbus’ eventful voyage. Once such scientist was a Greek mathematician by the name of Eratosthenes, who lived in the 2^{nd} century B.C.E. His experiment to determine the circumference of the Earth took for granted that the Earth was a sphere. He reasoned that because the sun cast no shadow at noon on a particular day of the year in one city, and it did cast a shadow at that same time in another city to the north, he would be able to measure the angular difference between the two shadows and use this information to determine how big around the Earth was. His calculations were simple and remarkably accurate.

- 2 meter sticks
- 2 protractors
- 2 buckets
- Potting soil or sand
- A watch
- A calculator
- A friend to help in a city on the same side of the equator as yours and at least 30 miles north or south of the city you live in. The city your friend lives in does not need to be directly north or south; it can be off to the east or west as well, but a city due east or west will not work for this experiment.
- Access to the internet

- You will need to choose a bright sunny day for both you and your partner.
- Use an internet resource to find out at what time the sun will be at its highest point on the day of your experiment. It will most likely not be at 12:00 noon.
- Fill the bucket with sand.
- Take your supplies out into the sun, where there are no shadows, about 20 minutes before the sun will be at its highest point.
- Place the meter stick in the bucket so that it is pointing straight up from the ground.
- Measure the length of the shadow the stick casts about 10 minutes before the sun will be at its highest point.
- Record this information on a chart such as the one below.
- Measure the length of the shadow again after about 2 minutes.
- Record the information.
- Continue in this fashion until the shadow stops shrinking as starts to grow.
- Use the shortest length of shadow for the rest of the calculations.
- Measure the height of the stick from the ground.
- Record this information.
- Divide the length of the shadow by the height of the stick. This is the tangent of the angle formed at the top of the stick.
- Use a scientific calculator (or a program on the internet) to find the inverse tangent (denoted as tan-1) of your answer to number 14. This is the angle formed between the top of the stick and the edge of the shadow.
- Your partner needs to complete steps 2-15 in their city on the same day. Have them send you the information through e-mail.
- Find out the distance between your city and the equator. Use a map or the internet to find this information.
- Find out the distance between your friend’s city and the equator. Use a map or the internet to find this information.
- Subtract the smaller distance from the larger one.
- Subtract the smaller angle from the larger angle.
- Because there are 360 degrees in a circle, you can use this number to set up a simple ratio to discover the circumference of the Earth.
- The ratio is (degrees difference or your answer in 20) / 360 = (distance between towns or your answer to 19) / the unknown circumference of the earth
- To solve the equation, you need to move the unknown quantity onto one side of the equation. Your new equation is 360 / (your answer to 20) X (your answer to 19) / 1
- Multiply 360 by your answer to 19 and divide by your answer to 20.
- Check to see if you answer is close to the actual circumference of the earth.

Time |
Distance of Shadow |

**Terms/Concepts:** Eratosthenes; Circumference;Angle; Noon; Zenith; Latitude; Longitude

**References:**