Measure the Size of the Earth
I propose the following challenge: that you measure the circumference of the Earth yourself.
Yes, you! And don't worry, you won't need to travel around the world with a tape measure 🤭
➤ A bit of history: Eratosthenes and his epic feat of measuring the Earth!
Imagine this: we're in ancient Greece, over 2000 years ago. There are no Google Maps, no satellites, nothing like that. But there's a guy named Eratosthenes who decides to figure out how big the Earth is. And he does it with amazing precision! How did he do it? Well, hold on, because this story is incredible.
Who was Eratosthenes?
Eratosthenes was a genius in every way. He was born in Cyrene (what is now Libya) and was an expert in everything: mathematics, astronomy, poetry, philosophy... he was even the head of the Library of Alexandria! Yes, the legendary library itself.
Eratosthenes' Trick
Eratosthenes made the calculation in a simple and ingenious way. Here's the step-by-step explanation:
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Summer Solstice in Syene (Aswan)
He knew that in Syene, a city in southern Egypt, at noon on the summer solstice, the sun was directly overhead. That is, no shadows! Objects looked like they had disappeared.
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Measuring Shadows in Alexandria
On the same day and at the same time, Eratosthenes was in Alexandria, which is north of Syene. Here, things were different: there were shadows! He measured the height of an obelisk and its shadow and calculated that the angle was 7.2 degrees.
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The Magic Angle
That 7.2-degree angle was like a detective's clue. Because if the sun is directly overhead in one place and makes an angle of 7.2 degrees in another, it means that those 7.2 degrees are a fraction of the Earth's full circle (360 degrees).
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The Distance Between the Cities
Eratosthenes knew that the distance between Syene and Alexandria was about 5000 stadia (a unit of measurement at the time).
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Calculating the Circumference
Using a simple proportion, he deduced that if 7.2 degrees were 5000 stadia, then 360 degrees would be 50 times that distance. He did the math and voila! The circumference of the Earth would be 250,000 stadia.
Converting this to kilometers, it turns out he was quite close to the actual figure we know today: about 40,000 kilometers. Not bad for not having a calculator, huh!
➤ Now it's your turn!
Would you like to follow in Eratosthenes' footsteps and measure the diameter of the Earth yourself? Here’s a guide to help you do it easily!
Date and Time
June 21st at 12 noon solar time
Materials
- Sun (to measure the shadow)
- Stick
- Tape measure
- Paper and pencil for notes
- Calculator
- Internet
- Level (optional)
Step-by-Step
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Longitude and Latitude
Find the longitude and latitude of the place where you will take the measurements. You can use Google Maps:
- Search for the location where you will take the measurements
- Right-click on your location
- Write down the longitude and latitude on your paper
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Clock Time Corresponding to Solar Noon
Nowadays, most time zones do not correspond to solar time, so you need to find the clock time that corresponds to solar noon. You can use the Global Monitoring Laboratory website:
- Enter your latitude and longitude in the Location field
- Enter June 21st in the Date field
- Write down the solar noon time on your paper
Now you know what time to take the measurements. In my case, I will have to do it as close as possible to 13:53:08.
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Distance from Your Location to the Tropic of Cancer
You can find the distance between your location and the Tropic of Cancer using Google Maps:
- Search for the Tropic of Cancer location on Google Maps, for this you need to search for the Tropic of Cancer's longitude which is 23.43806 and your location's latitude (in my case 2.17001)
- Right-click to measure distances
- Search for your location and click to measure
- Write down the distance on your paper (in my case 1995.82 km)
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Measuring shadows
Now it's time to measure the size of the Earth.
- You need to take the measurement on June 21st at solar noon (in my case at 13:53).
- Take your stick and place it in a spot where it gets sunlight so you can measure its shadow.
- Verify that the ground is completely horizontal; you can use a level for this.
- Make sure the stick is completely vertical.
- At solar noon, measure the shadow that your stick casts on the ground.
- Measure the height of the stick.
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Calculate the shadow angle
Use the measurements of the shadow and the height of the stick to calculate the shadow angle. You can use the trigonometric formula:
tan(angle) = length of the shadow / height of the object.- Take the calculator.
- Divide the length of the shadow by the height of the object.
- angle = arctan(length of the shadow / height of the object)
- Write down the result on your paper.
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Calculate the circumference of the Earth
Use Eratosthenes' proportion:
circumference = (360 degrees / angle difference) * distance between locations.
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Find the diameter of the Earth
Divide the obtained circumference by π (pi ≈ 3.14159) to get the diameter of the Earth:
diameter = circumference / π.
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Possible errors
The circumference of the Earth is 40,075 km and its diameter is about 12,756 km.
What values did you get? It's very likely that your values are different from the real ones, but they should be close.
It's very easy to make small errors that cause a deviation in the calculation. For instance, a single millimeter difference in the length of the shadow can result in a difference of over 100 km.
We also have to consider that the Earth is not perfectly round, that the measurement depends on the sea level we're at, that there is flattening of the Earth...
The important thing is that you get an approximate value and that you were able to calculate the diameter of the Earth using a stick. 👏👏👏
And that's it! With these simple steps, you can measure the diameter of our planet, just as Eratosthenes did over 2000 years ago. Good luck and have fun experimenting!