Measure the Earth’s Circumference with a Shadow
IntroductionHow long of a tape measure would you need to measure the circumference of the Earth? Would you need to walk the whole way around the Earth to measure it? Do you think you can do it with just a meter stick in one location? Try this project to find out! Important: this project will only work within about 2 weeks of the spring or fall equinox (usually about March 20^{th} and September 23^{rd}).
This activity is not appropriate for use as a science fair project. Good science fair projects have a stronger focus on controlling variables, taking accurate measurements, and analyzing data. To find a science fair project that is just right for you, browse our library of over 1,200 Science Fair Project Ideas or use the Topic Selection Wizard to get a personalized project recommendation.
BackgroundWhat is the circumference of the Earth? In the age of modern technology, this might seem like an easy question for scientists to answer with tools like satellites and GPS, and it might be even easier for you to look up the answer online. It might seem like it would be impossible for you to measure the circumference of the Earth using just a meter stick. However, the Greek mathematician Eratosthenes was able to estimate the circumference of the Earth over two thousand years ago, without the aid of any modern technology. How? Using a little knowledge about geometry! At the time, Eratosthenes was in the city of Alexandria in Egypt. He read that in a city named Syene south of Alexandria, on a particular day of the year at noon, the reflection of the Sun was visible at the bottom of a deep well. This meant that the Sun had to be directly overhead (another way to think about this is that perfectly vertical objects would cast no shadow). On that same day in Alexandria, a vertical object did cast a shadow. Using geometry*, this allowed him to calculate the circumference of the Earth using this equation:
In this project, you will do this calculation yourself by measuring the angle formed by a meter stick’s shadow at your location. You will need to do the experiment near the fall or spring equinox, when the Sun is directly overhead at the Earth’s equator. Then, you can look up the distance between your city and the equator, and use the same equation Eratosthenes did to calculate the circumference of the Earth. How close do you think your result will be to the “real” value? * There is a geometric rule about the angles formed by a line that intersects two parallel lines. Eratosthenes assumed that the Sun was far enough away from the Earth that its rays were effectively parallel when they arrived at the Earth. This told him that the angle of the shadow he measured in Alexandria was equal to the angle between Alexandria and Syene, measured at the center of the Earth. If this sounds confusing, don’t worry! It is much easier to visualize with a picture. See the references in the More to Explore section for some helpful diagrams and a more detailed explanation of the geometry involved. Materials
Preparation
Procedure
Extra: try repeating your experiment on different days before, on, and after the equinox, or at different times before, at, and after solar noon. How much does the accuracy of your answer change? Extra: ask a friend or family member in a different city to try the experiment on the same day and compare your results. Do you get the same answer? Observations and ResultsIn 200 B.C.E., Erastothenes estimated the circumference of the Earth to be about 46,250 km (at the time he used a different unit for distance, the stadia). Today we know that the Earth’s circumference is roughly 40,000 km (24,854 miles). Not bad for a twothousand year old estimate with no modern technology! Depending on the error in your measurements, like the exact day and time you did the experiment, how accurately you measured the angle or length of the shadow, and how accurately you measured the distance between your city and the equator, you should be able to calculate a value fairly close to 40,000 km (within a few hundred, or maybe a few thousand, km). All without leaving your own back yard! More to Explore
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CreditsBen Finio, PhD, Science Buddies
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Key Concepts
Circumference, angles, parallel lines, Earth’s equator, latitude

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