Let's measure our planet
Recreate Eratosthenes' famous experiment!
Can you believe that more than two-thousand years ago, one man was so curious and surprised by the behaviour of shadows that he was able to measure the circumference of our planet?! This page will give you everything you need to work through these same experiments with your students.
We’ve created videos that explain everything in simple, visual terms. You can also bring these into your classroom! You can view the whole playlist here, or find each video within its section below.
Explicatory videos will be made available in the coming weeks.
Get Started!
You can use the shadows cast by the equinoctial Sun to recreate Eratosthenes’ legendary experiment to measure the circumference of the Earth. With just a couple of sticks and a measuring tape, this easy-to-replicate experiment gives you the opportunity to combine lessons to create an interdisciplinary learning experience across astronomy, mathematics, geography, and history! (Don’t worry, if math isn’t something you want to dive into too deeply, we have developed a simple calculator you can use instead.)
You might want to consider participating in the international Eratosthenes Experiment to calculate the Earth’s circumference, where they’ll pair your class with another school somewhere else in the world!
Historical Context
First, let’s meet the mind behind the experiment! Eratosthenes was a Greek mathematician, astronomer, geographer, and chief librarian of the Library of Alexandria. He loved combining careful observations with clever thinking to solve problems that seemed impossible. Around 240 BCE in Ptolemaic Egypt, he measured the circumference of the Earth with 95% accuracy —that’s no small feat!
While reading at the library, Eratosthenes came across an account describing that no shadow was cast at the bottom of the wells at noon on the summer solstice in Syene. He found this was not the case in Alexandria, where shadows were cast at the same local time. Using his knowledge of geometry, he measured the angle between the shadows and the sun's rays in Alexandria to be about 7.2 degrees at that precise time.
Image credit: ASTROLab du Mont-Mégantic
By following his curiosity, Eratosthenes not only correctly identified the Earth’s curvature, he also formulated how his simple measurements could be used to extrapolate the circumference of the entire Earth. To his reasoning: if in Alexandria, at some distance north of Syene, a pillar casts a shadow of about 7.2 degrees —roughly 1/50th of 360 degrees, the angle of a full circle— then surely the distance between the cities must also be 1/50th of the spherical Earth’s circumference. Multiplying by 50 would then yield the circumference of the Earth.
Back then, professional step-measurers would pace along roadways to measure distances between cities. Thanks to them, Eratosthenes had access to the number he needed: the distance between Alexandria and Syene, in stadia, a now-obsolete unit. Eratosthenes’ calculations yielded the circumference of Earth as 252,000 stadia which is 98% accurate to the currently accepted value of the Earth’s circumference (40 008 km through the poles). Using Eratosthenes’ masterful methodology, and with access to more reliable distances between cities, Earth’s circumference can be calculated even more accurately by you and your students!
Preparation
Participating in the Eratosthenes Experiment
If you’d like to participate in the worldwide initiative that recreates this legendary experience, submissions open for the Eratosthenes Experiment twice a year around each equinox. To participate, you will need to make an account on https://eratosthenes.ea.gr/ and follow the instructions to submit your measurements. They also have a photo contest every equinox that you can participate in with your class!
Finding your date and time
For practical reasons, it is easier to do the experiment on the equinox. On those days, the Sun is directly over the equator, making the calculations easier. Also, equinoxes are the dates chosen for the international program, if you’d like to collaborate with another school. So select a date close to March 21 or September 22. When the Sun reaches its peak for the day, it’s known as solar noon —but it’s rarely actually at 12-noon! So you’ll need to find solar noon for your location (resources below).
At the equator, the Sun’s trajectory passes precisely overhead on equinox, which means objects do not cast shadows (this is important later). But, did you know that the Sun is actually never directly overhead in Canada? For us, the Sun’s path is always a bit tilted toward the south. So as you work with the Sun, even at its highest point in the sky, you’ll always find a shadow. Subsequently, shadows will be cast north.
When the Sun reaches its highest point on equinox, the length of shadows cast is determined solely by the height of the Sun in the sky. Note: If you measure the shadow earlier or later in the day, the shadow will be longer. This is because, with the rotation of the Earth, the Sun appears to move up in the morning and down in the afternoon.
Here’s how to find your best time for running the experiment:
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Choose a date: Ideally, you would run your experiment on the day of the equinox, but a few days before or after won’t affect the results much.
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Check the weather: You also need to consider the weather: clear skies work best, but as long as you can manage to get sunlight bright enough to cast a measurable shadow, you should be good to proceed!
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Find solar noon: The timing of solar noon depends on your position in the time zone, whether daylight savings is in effect, and a few other factors. To find your solar local noon, you can:
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Look at sites such as timeanddate.com or NOAA Solar Calculator.
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Use Stellarium Web for your location and adjust the date and time until the Sun is exactly due south.
Gather the materials
To perform the experiment, all you need is a:
A “stick,” a straight and narrow object that you can stand upright, perpendicular to the ground, will work. We suggest using a metre stick, a plunger, a pipe, or a broom! Short objects, like pens or pencils, may be too difficult to measure with precision. Ideally, the object can stay upright on its own, but you can also have someone hold it steady.
A measuring tape or a ruler. Units of the measurement won’t matter as long as you are consistent about using the same unit of measurement when measuring both the lengths of the stick and its shadow.
Optional: You may consider using a white sheet or other flat surface on which to cast the stick’s shadow. Bumpy or irregular ground, like a patch of grass, may make seeing and measuring the shadow a challenge.
Make your measurements
First, measure the length of your stick. Then, before solar noon, set the stick upright on flat ground, if using a sheet or other screen for the shadow, set this up as well. Make sure your stick stays upright and casts a clear, measurable shadow.
At solar noon, measure the length of the shadow from the base of the stick to the tip of the shadow, and record your measurement.
Now you are ready to make your calculations. Check the mathematics section below or use our calculator to make it easier for you and your students.
Mathematics
Here we dive into the geometry and mathematics behind the experiment. If that’s not something you wish to get into, you can skip this portion by using our simple calculator.
Background in Geometry
You may want to discuss the geometry of this experiment with your students. As shown in the diagrams below, parallel light rays from the Sun are responsible for casting a stick’s shadow. Due to the curvature of the Earth, a ray of light that casts no shadow on the equator, will cast a shadow away from the equator.
Since the beam of sunlight falling on the Earth is a composite of parallel rays, you can use this experiment to discuss the properties of parallel lines.
Why do we only measure the angle between the tip of the stick or pillar and the sun’s ray?
This is because parallel lines create equal alternate interior angles. This angle at the centre of the Earth is a fraction of a full circle. For Eratosthenes in Alexandria, this angle was 1/50th of a full circle. So, the part of the Earth’s surface corresponding to this angle —the distance between Alexandria and Syene— was also 1/50th of the Earth’s circumference.
You could use this opportunity to discuss further topics related to circles. Would you see this same phenomenon if the Earth was not spherical? No, because a parallel beam of sunlight on a perfectly flat Earth would cast shadows of the same length in any two cities on Earth!
Calculate: Measure the Earth
Step 1: Find the value of the angle that represents the height of the Sun in the sky (the angle θ in the diagram below).
To do this, we cast a shadow with a vertical stick and compare the length of the stick to the length of its shadow. There are different methods to get the value of that angle but do this by measuring the length of a stick and its shadow, explained below. You could also draw a triangle similar to the one created by the stick and its shadow and measure the angle with a protractor; use an app that will allow you to measure angles directly through the camera; or get creative!
Finding the angle θ: First calculate the ratio of the length of the stick and the length of the shadow. This gives us the tangent of the angle the stick makes with the sun’s rays.
To get the angle, you need to calculate the inverse tangent or arctan of this ratio, with this equation: θ = arctan (S/H)
In other words, using a calculator you will:
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Divide the shadow length by the stick height.
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Press the arctan (or tan⁻¹) button on a calculator (depending on your calculator, you may need to insert the above value).
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That's your angle θ.
The value your calculator returns is the angle we need to recreate the Eratosthenes experiment! Note: If your calculator’s default angle is set to radians, you may have to change that setting to degrees, or manually multiply the answer by 360/π to get the angle in degrees.
Length of shadow / Height of stick = S/H = tan(θ).
Step 2: Find this angle for a different location.
To carry out Eratosthenes’ experiment and calculate the circumference of the Earth, you will now need to acquire the angle from a different location along your longitude. This is where you might want to partner up with another school, or you can assume a fictional school at the equator, for which the angle would simply be zero (like Syene on the solstice, there’d be no shadows cast there!).
Visual representation of the angles.
A full circle is divided into 360°. This means that any angle measured at the centre of a circle corresponds to a proportional distance along its circumference. Because the two cities lie along the same longitude, the distance between them follows the curve of Earth and represents a fraction of Earth’s total circumference. The fraction of Earth’s circumference (distance) between the two cities is the same fraction of the full 360° circle represented by the angle at Earth’s centre.
Step 3: Calculate the difference between the angle at your school (School A) and the angle from a sister school along your longitude (School B), measured at the same time.
Difference in angles = Angle at School A − Angle at School B
If both schools are in the same hemisphere, subtract the smaller angle from the larger angle. The direction (north or south) may make the result positive or negative, but for this calculation you only need the value of the difference.
If the two schools are in opposite hemispheres, add the two angles together to find the total difference between the locations.
Eratosthenes used a similar approach when comparing Alexandria with Syene. At solar noon in Syene, the Sun was directly overhead, meaning the shadow angle was approximately 0°. As a result, the difference between the two angles was simply the angle measured in Alexandria. You can also use 0 degrees, as if partnering with a fictional school on the equator!
Step 4: Find the distance between your school and the other school (or the equator).
There are online tools that allow us to measure the longitudinal distance between places on the surface of the Earth. Ideally, the two points are on the same longitude.
This is a great opportunity to include some geography, explore online tools and learn more about our planet at the same time!
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Google Earth: This is our preferred measuring tool. Using the ruler icon in the toolbar (top), you’ll click on your school’s location, and then find your second location. Here’s a quick tutorial: Measuring with GoogleEarth
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National Geographic MapMaker allows you to measure distances using a flat map. Use the Measure option in the toolbar (bottom), and select the two points directly on the map. You can zoom in to help you. For the equator, go to a latitude of 0 with a longitude as close as possible to yours.
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NOAA Calculator: Simply enter the latitude and longitude of your two points. For the equator, use a latitude of 0 and the same longitude as your school. This option is simplest but doesn’t give a visual overview of what students are measuring. It can also be used to double-check the other values measured on the map.
Step 5: Now that you have all the values, you can do the final calculation and get the result you were waiting for: the circumference of our planet!
Earth’s Circumference = (360 / Difference between Angles) x (Distance between School A and School B)
What did you get for the circumference of the Earth? Do you have a value close to the real circumference of 40,008 km?
We use Earth’s polar circumference (40, 008 km) as the reference rather than its equatorial circumference. Since this experiment measures the distance along a line of longitude, it corresponds to the polar circumference. Did you know? The Earth is not a perfect sphere, it’s actually slightly 'squashed' at the poles. So these two measurements are different. The polar circumference is about 67 km shorter than the equatorial circumference.
Discussion on Calculations
The Eratosthenes experiment can be used in a classroom setting to explain or introduce ratios, particularly trigonometric ratios, as well as inverse trigonometric functions. For students who are not yet exposed to trigonometry, you can use this opportunity to talk about fractions, and simply use a calculator on your end to calculate the inverse tangent for the class as a collective. This is also a good opportunity to talk about measurement methods and unit conversions —if you don’t measure both the stick and the shadow in the same unit (say, centimetres or inches), you won’t get the right answer! Or if the value your calculator gives you is in radians, then it won’t be a fraction of 360, but of 2π! That being said, the distance between School A and School B can be in any unit you prefer because there are no other physical quantities in the equation for that step that might cause a mismatch of units. The answer for the circumference will be in those units. Eratosthenes used stadia, but we can use kilometres, meters, or even miles, depending on what is the most accessible and makes the most sense for the class.
Frequently Asked Questions
classroom-ready presentation
We’ve created ready-to-use slide decks to make sharing these themes with your students easy and engaging. Just download the Google Slides presentation (in whichever format you prefer), then customize anything you’d like so it fits your teaching style and goals.
Resources & IMages
Here are some resources to help you as you plan this experiment for your class:
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Explanation of the Eratosthenes Experiment by Carl Sagan (6:41 video - great to show to your students!)
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Our Google Sheet calculator for easy math
Below, you can download all our images relevant to these activities to use as you wish. If you're sharing something of your own, please credit Discover the Universe and link back to this webpage.
SUPPORT
We think the Eratosthenes experiment is a great way to bring history, mathematics, and science together in your classroom. After reading through these steps, we hope you’re inspired to use it as a backbone for cross-curricular activities.
Should you run into any trouble, or need advice on anything, please feel free to reach out to our team and we'll get back to you with support as soon as we can!