Measure the Earth’s Circumference with a Shadow
Summary
Introduction
How 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 20th and September 23rd).
Background
What 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:
- He knew there were 360 degrees in a circle.
- He could measure the angle of the shadow cast by a tall object in Alexandria.
- He knew the overland distance between Alexandria and Syene (the two cities were close enough that at the time, this distance could be measured on foot).
- The only unknown in the equation is the circumference of the Earth!
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
- Sunny day on or near the spring or fall equinox (about March 20th or September 23rd)
- Flat, level ground that will be in direct sunlight around noon
- Meter stick
- Volunteer to help hold the meter stick while you take measurements. If you are doing the experiment alone, you will need a bucket of sand or dirt to insert one end of the meter stick to hold it upright.
- Optional: plumb bob (you can make one yourself by tying a small weight to the end of a string) or post level to make sure the meter stick is vertical
- Calculator
- Two options to measure the angle formed by the shadow:
- A protractor and a long piece of string
- A calculator with trigonometric functions
Preparation
- Look at your local weather forecast a few days in advance, and pick a day where it looks like it will be mostly sunny around noon. You have a window of several weeks to do this project, so don’t get discouraged if it turns out to be cloudy! You can try again.
- Look up the sunrise and sunrise times for that day in your local paper or on a weather website. You will need to calculate “solar noon,” the time exactly halfway between sunrise and sunset when the sun is directly overhead. This will probably not be exactly 12:00 noon.
- Go outside and set up for your experiment about 10 minutes before solar noon so you have everything ready.
Instructions
- Set up your meter stick vertically, outside in a sunny spot at solar noon.
- If you have a volunteer to help, have them hold the meterstick. Otherwise, bury one end of the meter stick in a bucket of sand or dirt so it stays upright.
- If you have a post level or plumb bob, use it to make sure the meter stick is perfectly vertical. Otherwise, do your best to eyeball it.
- Mark the end of the meter stick’s shadow on the ground with a stick or a rock.
- Draw an imaginary line between the top of the meter stick and the tip of its shadow. Your goal is to measure the angle between this line and the meter stick. There are two different ways to do this:
- Method 1: use a protractor
- Have your volunteer stretch a piece of string between the top of the meter stick and the end of its shadow.
- Use a protractor to measure the angle between the string and the meter stick in degrees. Write this angle down.
- Method 2: use a calculator and trigonometry
- Mark the exact spot where the meter stick touches the ground (or where it would touch the ground if it wasn’t inside a bucket).
- Use the meter stick to measure the length of the shadow (the distance between the bottom of the meter stick and the tip of the shadow that you marked previously). Write this distance down (in meters).
- Use a calculator with trigonometric functions to calculate the angle, using the atan (or tan-1) button. Make sure the calculator is set to degrees and not radians. Then enter atan(distance), where “distance” is the length of the shadow you measured in the previous step. Write the resulting angle down.
- Note: the previous step assumes you are using a meter stick and not a yard stick. If you are using a yard stick, measure the shadow length in inches, and then calculate atan(distance/36).
- Method 1: use a protractor
- Look up the distance between your city and the equator. There are several options to do this:
- If you have internet access, there are websites like https://www.timeanddate.com/worldclock/distances.html?n=12 that will automatically calculate the distance between a city and other locations (other cities, the north and south poles, the equator, etc.).
- Use the scale on a world map to measure the distance between your city and the equator. Note that since flat maps always require some sort of distortion, this method may not be very accurate for cities that are far away from the equator.
- Here is a list of some major US cities and their distance from the equator. If you are at the same latitude as one of these cities, you can just use these values. Or, you can look on a map of the United States to determine how far north or south you are from one of these cities, and add (or subtract) that value from the distance:
- Miami, FL: 2,852 km (1,772 mi)
- Orlando, FL: 3,159 km (1,963 mi)
- Houston, TX: 3,293 km (2,046 mi)
- San Diego, CA: 3,621 km (2,250 mi)
- Dallas, TX: 3,629 km (2,255 mi)
- Phoenix, AZ: 3,703 km (2,301 mi)
- Atlanta, GA: 3,737 km (2,322 mi)
- Los Angeles, CA: 3,769 (2,342 mi)
- Charlotte, NC: 3,899 km (2,423 mi)
- Nashville, TN: 4,004 km (2,488 mi)
- San Francisco, CA: 4,183 km (2,599 mi)
- Washington, DC: 4,308 km (2,677 mi)
- Kansas City, MO: 4,329 km (2,690 mi)
- Baltimore, MD: 4,350 km (2,703 mi)
- Denver, CO: 4,402 km (2,735 mi)
- Philadelphia, PA: 4,424 km (2,749 mi)
- New York, NY: 4,508 km (2,801 mi)
- Chicago, IN: 4,637 km (2,881 mi)
- Detroit, MI: 4,688 km (2,913 mi)
- Boston, MA: 4,691 km (2,915 mi)
- Portland, ME: 4,836 km (3,005 mi)
- Minneapolis, MN: 4,983 km (3,096 mi)
- Portland, OR: 5,042 km (3,133 mi)
- Seattle, WA: 5,277 km (3,279 mi)
- Calculate the circumference of the Earth using this equation:
What value do you get? How close is your answer to the true circumference of the Earth (see Observations and results section)?
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 Results
In 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 two-thousand 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!
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Additional Resources
- Calculating the Circumference of the Earth, from Science Buddies
- Lesson: Measure the Earth's Circumference, from ASEE
- Angles, parallel lines and transversals, from Math Planet
- Transversal (geometry), from Wikipedia
- Science Activities for All Ages!, from Science Buddies
