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The activity allows you to measure the radius and circumference of Earth via simple experiments that strongly rely on scientific collaboration between distant locations and schools. Additional large-scale parameters such as the tilt of Earth’s axis, astronomical noon and Earth’s position around the sun can be measured with the complete set of experiments.
The project explores various additional phenomena that can be linked to topics in physics, astronomy, geography, history and cultural awareness. The results can be analysed using simple geometry, or more sophisticated mathematics, resulting in an activity suitable for a wide range of ages.
Astronomy, Physics, Geometry, Math
Scroll down to read tips for educators.
In this project you will explore how Earth’s rotation around the sun can be studied and measured by observing the shadows projected by a known object. You will learn how geometry and math can be used to measure very large objects, sometimes without even touching them!
You will also experience the strength of scientific collaboration. By working together with participants in different areas of New Zealand, you will experience how the same measurement is affected by the location, and how each one of us can contribute to a grander objective, only achievable by working together and sharing our knowledge.
In the end, we will combine the measurements you collected from your unique spot in New Zealand with those from other participants across the country, and achieve something that none of us would be able to do by ourselves: we will measure the size of our planet!
Earth’s radius and circumference:
Sunlight projection:
More than two thousand years ago, there was a Greek polymath (a person who studies many different things) named Eratosthenes, who became the first person to estimate the size of our planet, Earth.
Eratosthenes knew (or maybe heard) of a very interesting well in the city of Syene (currently called Aswan, in Egypt). This well was so deep, and so narrow that light from the Sun could not reach its bottom, so no one could see how deep it was or how much water it had. However, once a year, at noon on the day of the summer solstice, the Sun’s reflection could clearly be seen on the surface of the water, deep in the well. This meant that sunlight was reaching the Earth completely vertical, so the edges of the well did not produce any shadows on the bottom.
Eratosthenes also knew that on the same day and at the same time, an obelisk (a type of tower) located in Alexandria, about 800 km north of Syene, did cast a shadow that could be observed and measured.
He realised that this difference was caused by the curvature of the Earth surface, as shown in Figure 1. He also noticed that the length of the shadows was related to the angle with which sunlight reaches the surface of the Earth, and that this angle can be used to estimate the size of the planet.
Eratosthenes came up with this great idea: he would place two poles perpendicular to the ground: one in Syene, and one in Alexandria. At noon on the day of the summer solstice, the pole located in Syene would not produce a shadow (just like the well he knew of). On the other hand, the pole placed in Alexandria would produce a shadow that he could measure, and use its length to eventually find the angle of the sunlight. This is shown in Figure 2:
Eratosthenes measured the difference in the angle of sunlight between the two cities, and it was about 7.2°. He also knew that the angle swept when drawing a complete circle is 360°. So the angle he had measured was 50 times smaller than the angle of a full circle. We can express this mathematically as:
Finally, he learnt from caravans of traders between Alexandria and Syene that the distance between these two cities was approximately 924 km.
So Erathostenes thought: “If the angle corresponding to a full circle is 50 times the angle I measured between the two cities, then the distance of a full circle (the perimeter) should be 50 times the distance between the two cities. So he calculated the perimeter of the Earth as:
Erathostenes did this simple experiment more than two thousand years ago, before there were spaceships, satellites, phones, Interner or even electricity. The amazing thing about his experiments was that not only it was the first time that someone had applied the scientific method and systematic experiments to measure the size of the planet, but he was off only by 6000 km!!! This means that he could measure the size of the Earth within 15% of error, using only sticks and the sun!!!
Wouldn’t it be great if we could work together and measure the size of the Earth right here, from New Zealand? The problem for us is that there is no place in New Zealand where the sun is directly above us on a particular day, like it is in Syene (as a challenge, I will leave it to you to think of why there is no place in New Zealand where the sun is directly above us). So we cannot exactly reproduce what Eratosthenes did.
However, we have a long enough country, full of smart young people who surely will be able to solve this by working together!
Here is how we will do it:
You will need to measure the shadow of a vertical pole during the day (and ideally during several days), and record the length of the shadow and the length of the pole. This will allow you to calculate the angle of the incident sunlight at different times of the day. As a minimum, we will need a single measurement taken at noon on multiple days.
You will need to compare your measurements with those from other participants, so it is a good idea we agree on what days and times to take the measurements before we start. Ideally you would take measurements, say every hour or two, from 9am to 5pm. This will allow us to study many other astronomical features and phenomena!
The idea is that two participants separated by a north-south distance d (we can measure this in a map), doing the experiment at the same time, will measure different lengths of shadows and different sunlight angle. By combining the measurements done by multiple participants at the same time of the day, we will be able to determine the size of our planet!
Maths curiosity: Don’t worry if the poles or stick we use are different, the angle of the sunlight is what we care about and this will not depend on the length of the stick.
Take a look at Figure 3. You and another participant are represented by the points A and B. The shadows and angles θA and θB correspond to measurements at points A and B.
Geometry fact: Did you notice that the angles measured from each side of line of the sticks to the horizontal sunlight are the
same at the centre of the earth and at the end of the stick? These are called alternate interior angles, and they are always the same.
So how will we measure the size of the Earth?
Once you have your measurements, you can share them with the rest of the participants. Each one of us will be able to calculate the perimeter and radius of the Earth, similarly to what Eratosthenes did.
Have a look at Figure 4 to guide you in the process:
Take the difference between your angle and the one measured by a participant in a distant location at the same time. At this stage it is a good idea to check that we have calculated our angles correctly.
Back to the measurement: knowing the distance between the two participants (you can check this online in maps.google.com), and the difference in angles (see Figure 4) you will be able to calculate the perimeter of the Earth, using the same relationships Eratosthenes used (Equations 1 and 2). In a nutshell: you divide the difference in angles by 360° and you multiply the result by the distance between the places where the measurements took place.
Once you have the perimeter of Earth, you can obtain the radius of the planet using:
So rearranging, the radius results:
Do not worry if the math seems too complicated at this stage, it will all become clear when you do the experiment, and there will be plenty of help available to assist you with the calculation.
A final note:
If you noticed that someone made a mistake on their measurements, it would be nice to politely point it out. Feel free to comment, ask questions and discuss this and anything else on the online forum of the project. Participants and scientists will be there to help and discover new things together. This way we will be helping everyone with their experiments.
1. Find an open space with flat ground.
2. Fix the straight stick on the ground or ask a friend to hold it vertically while you check the alignment and take measurements.
3. Here is an example of how your experimental setup might look like:
4. Using the level or plumb bob, make sure that the stick is perfectly vertical.
5. Using the measuring tape or metre stick, measure the height of the stick and the length of the shadow it casts.
6. Repeat these measurements as often as you can. Ideally every hour or so, for a few days.
To think and discuss: Can you identify a time of the day when the shadow is the shortest? And the longest? Why do you think this happens?
7. Use your results to determine the angle of the incident sunlight at different times of the day (especially at noon). You can find two methods for doing this in the section below.
8. Share your results with other participants experimenting far away from you, and use the method described in the previous section to estimate the circumference and radius of the Earth. You can always get in touch with other participants and with your friendly scientists to get help with this or to discuss your results. Use the forum on our website to get and to offer help and to learn more about your place in the universe!
Method 1 (graphic method)
Once you have measured the length of the shadow and the length of the pole or stick, you can determine the angle of the incident sunlight by drawing your experiment to scale and measuring the angle with a protractor.
Let’s say that the length of our stick is 1 m (100 cm) and the length of the shadow we measured is 67 cm. We can draw our experiment to scale, making 1 cm in the drawing represent 10 cm in the real world. The stick in our drawing will then be 10 cm long, and the shadow will be 6.7 cm long. Remember that these two will be perpendicular to each other (the ground is horizontal and our stick was perfectly vertical). The drawing will then look something like this:
Since we drew our experimental setup to scale, the angle in the drawing will be the same as the one in the real experiment (the angle formed by sunlight and the vertical pole). We can then use a protractor to measure the angle like this:
In our case, the angle of the incident sunlight turned out to be 33.5°.
Method 2 (using trigonometry)
Alternatively, if you really like math you can calculate the angle using trigonometry.
The angle is related to the ratio between the length of the shadow and the length of the stick. You can calculate this ratio and then use the inverse tangent to get the angle:
In our example, we will get:
To think and discuss: Which method do you think is better to determine the angle? Why?
Let’s see if we can visualise in a simple physical model what is going on at a larger scale.
1. Take the piece of cardboard the size of an A4 sheet or larger, and measure two parallel sides.
2. Find the centre of each side and draw a line between them. The line should be run across the cardboard.
Math curiosity: is the line going through the centre of the cardboard? How can you find the centre? Will it be the same if the cardboard is square, rectangular or a parallelogram?
3. Starting from the middle of the line you just traced, draw marks every 10 cm along the line.
4. Place a straw on each mark and glue it to the cardboard using blu- tack or tape. Use the square to make sure the straw is perpendicular to the plane of the cardboard.
5. What you just made is a representation of the surface of the Earth. Each mark represents one of the participants collaborating with you. The straws represent the sticks you and others will be using to measure the size of the planet.
6. Take the cardboard to the sun and point it so that the surface is perpendicular to the light. Try to keep the cardboard as flat as possible. Do the straws cast shadows?
7. Slowly start curving the cardboard and see what happens to the shadows. Are they all the same length and orientation? Do the shadows change if you change how much you curve the cardboard?
8. Based on the model you created, do you expect the shadows measured by participants in different locations across New Zealand to be the same? Why?
9. If you have a globe at home or at school, you can reproduce the experiment on a spherical representation of the Earth. You
can also use a ball (although a rugby one will probably not be a very good model of the Earth).
10. Stick straws perpendicular to the surface of the ball and take the ball to the sun. Look at the different lengths and directions of the shadows. Do they change when you rotate the ball? Can you compare this with what happens on Earth?
Telling time: tracing the daily apparent path of the sun
1. Use the compass to identify the magnetic north pole and record its angle with respect to the shadow.
2. If you plan to take measurements several times a day and for a few days, it is a great idea to mark the position of the stick and the end of the shadow on the ground for each measurement. You can use chalk, painter’s tape or any other way of identifying the spots on the ground. Remember to write down the date and time of each measurement.
3. Repeat the measurements as frequently as possible over a few days or weeks and see if the length and orientation of the shadow
remain the same at the same time of the day. Would you be able to tell time based on the orientation of the shadow?
4. Can you imagine how to construct a solar clock using your observations? Give it a try!
The seasons: tracing the annual apparent path of the sun
1. Glue a 15cm long stick on a piece of cardboard and use a square to make sure it remains perpendicular to the surface.
2. Orient the cardboard with one side aligned to the magnetic North pole and paint a dot on the cardboard, where the shadow ends.
3. Label the dot with the date and time it was painted.
4. Repeat this every hour over the first two days and then once a day (at the same time of the day) over a few weeks and once a week over the next few months.
5. At the end of the first day or two, you will observe a pattern. Can you propose an explanation for this?
6. Over a few weeks and months, a new pattern will arise. Can you imagine what this corresponds to?
7. If you can see the sun from your window, you can also use a permanent marker to mark the position of the sun once a week, always at the same time. Just draw a dot on the window where the sun is. (Don’t worry, you can remove the marks on the window later with alcohol).
8. If you do this every week at the same time throughout the year, you will see a very particular pattern emerging. Just be patient (and constant) and give it a try!
Beyond Eratosthenes: using the sun to measure tall objects
1. Think of how you could use the tools and methods you used here to measure the height of tall objects like trees or houses.
2. Consider measuring the shadow of an object and try to use the methods you used in this project to determine the height of a friend, a tree and a building.
3. What are the most relevant parameters/variables to consider when doing these measurements?
4. Discuss the benefits and limitations of these kind of measurements.
Watch our example of an experiment set up:
Head to the forum to find out when we’ll start running this experiment. In the meantime, you can get your materials together.
Our solar system is big, really BIG!!!
It is so big that using kilometers to measure its size is not really very efficient. The numbers become too large.
For example, the diameter of our sun (we call it Sol) is approximately 1392700 km (that is around 1.3 million kilometers). The closest planet, Mercury has a diameter of approximately 4880 km and it orbits at approximately 58000000 km from the sun. So between the surface of the sun and the first planet, you could put more than 40 suns!
The Earth orbits the sun at a distance of around 150000000 km. You can already see the numbers getting large!
Awesome fact: light travels in the vacuum of space at approximately 300000 km/s. This means that light emitted by the sun right now will reach us in a little bit more than 8 minutes. So by the time you finish reading this, the light emitted when you started would not have reached us yet!
The planet orbiting the furthest away from the sun is Neptune. Its distance from the sun is approximately 4470000000 (4.47 billion kilometers). It takes about 4 hours for light from sun to reach Neptune!
Sizes, distances and big numbers
Here is an easy experiment to try:
1. Find out what is the largest planet in the solar system. Try doing a little research and write down its diameter and the distance to the sun.
2. Find out what is the smallest planet in the solar system. Write down its diameter and the distance to the sun.
3. Try drawing the sun, the smallest planet and the largest planet in the solar system, to scale.
For example, we can represent 1 million km by 10 cm. Our scale factor would then be 10 cm/million km.
We know that the sun is 1.5 million kilometers in diameter, so using our scale factor, we would draw the sun as a circle of 15cm diameter (1.5 million km × 10 cm/million km)
We mentioned above that Mercury’s diameter is 4880 km. How big would the circle representing Mercury be?
Try doing this with the smallest planet and the largest planet in the solar system. Do you see a problem?
Imagine that we now want to draw the planets, one next to the other, but respecting the distances from the sun. Keeping all to scale.
How far apart should they be on the drawing? a few millimeters? a few centimeters?
Technology to the rescue!
Our BIG SCIENCE! scientist created this Augmented Reality environments to overcome the problems of large distances, of large numbers and and of cloudy days.
Here you have our solar system represented using different scales. That way we can appreciate the relative sizes of the planets, while tracing their trajectories around the sun.
Our scientists are so powerful that they managed to speed up time itself, so you can explore different aspects of the solar system without having to wait long times for the planets to ‘move’ on the real sky. We managed to speed up time, for all planets, so that an Earth’s year in this AR environment takes only a few seconds.
Note: if you have trouble visualising our AR in your mobile device, try using your device in landscape mode, or changing the visualisation mode to ‘desktop site’ on your browser’s settings.
AR experiment 1:
a. Project the solar system on your backyard, your room, or your hand!
b. Take photos of cool o funny stuff going on in or around the solar system and share it in our forum.
c. Explore the solar system and try to identify each planet. Check out how fast the planets near the sun move, relative to those far away. Can you come up with a rule of their orbital period (the time it takes for a planet to complete a turn around the sun).
AR experiment 2:
a. Find Earth in the solar system
b. Measure what its orbital period is. Can you think of multiple ways of doing this? What would be the best method? How would you do it in the real world?
AR experiment 3 (the challenge):
We know that one year in Earth is different from a year in Saturn, Jupiter or Venus.
The challenge for you is this:
Can you think of a way of measuring a year in different planets, relative to Earth. In other words: how long, in Earth’s years, is a year in each of the planets of the solar system?
Think about these questions and see if you can start answering them, share your thoughts on the forum and get your calculations ready. Soon we will open the submission page for this part of the project, so we can find together YOUR PLACE IN THE UNIVERSE
How does science works?
Science in history and cultural awareness
The tools of science
Mathematics as a language for doing science
Relationships between real-world magnitudes represented in symbols. Computing one of magnitudes (symbols) by substituting all the others with the corresponding numerical values.
Distances and angles that are inconvenient to measure directly can be found from measurable counterparts, using scale drawings or formulas.
Modelling as a way to find a mathematical relationship that describes the objects or processes under investigation. A mathematical model may give insight about how something really works or may fit observations very well even when intuition fails.
There are a few assumptions we are considering in this project, which might be worth keeping in mind.
Depending on the level and time available, it might be worth discussing these assumptions with the participants. This could open the door to discussions over the strength of the models and the experiments in science, the limitations of different methods, and the importance of using adequate approximations, hypothesis and models to simplify the experiments and still
get accurate results.
Here are some of the assumptions:
A few other things you can discuss with your learners:
Here is an online simulator to calculate the results based on theory. You can use this to see what sort of values to expect from the experiments, find local noon, simulate the apparent trajectory of the sun or discuss things further with your learners.
If your students are keen on using this, there is even a free app for android! Search for “SunCalc org” in Google play store.
Check out how schools around the globe are working on similar, collaborative projects.
Check out this video of the measurement taking place in the original cities of Syene and Alexandria.
If you have any questions/comments or requests, feel free to get in touch with Dr Rodrigo Martinez Gazoni or any of the BIG SCIENCE! team
