How to Calculate the Circumference of the Earth

It all began in Greece in 240 BCE when a Greek mathematician and astronomer named Eratosthenes noticed something peculiar. He observed that on June 21, the day of the Summer Solstice, a stick placed vertically upright in the city of Syene in Egypt cast absolutely no shadow at noon.

This ignited Eratosthenes’ curiosity, making him wonder if the same would be true in Alexandria, the city where he lived. So, the next year, he carried out the same experiment, but in Alexandria.

The question was, would the stick again cast no shadow?

To his amazement, this time there was a shadow, and the angle made by the sun’s rays with the top of the stick (Sun angle) was 7.2°.

Source for all images in this post: Business Insider

Now, Eratosthenes was in a serious dilemma. If the Earth was flat– as was contemporary belief– and the light rays from the sun were parallel, then how was it possible that the same stick cast a shadow at Alexandria but not at Syene?

This was only possible if the Earth was spherical, as hypothesized earlier by Aristotle and Pythagoras. So, Eratosthenes concluded that the Earth is a sphere. But he didn’t stop there– he wanted to calculate the circumference of this sphere.

Eratosthenes already knew the Sun angles at Alexandria and Syene to be 7.2° and 0° respectively. From this information, he could infer that Alexandria and Syene were 7.2° apart on the Earth’s 360° surface.

Now, all he needed was the distance between Alexandria and Syene so that he could apply the concept of basic proportions and ratios to calculate the circumference of Earth. So, he hired a man to walk all the way from Alexandria to Syene and learned that they were around 5000 stadia or 800 km apart.

Today, we know the true circumference of the Earth is 40,075 km. And just like that, using just a stick and his brain, a man who lived 2,200 years ago found the circumference of our entire planet with over 99% accuracy.

And it turns out that you can do it too by performing this following experiment!*

  1. Check for your local noon time on the Internet (the local noon time is when the sun is at its zenith in the sky).
  2. Take a stick (or any other vertical object), place it upright, and record the length of its shadow at the local noon time.
  3. Now in order to determine the Sun angle, draw a right-angled triangle with the length of the stick (or object) as the base and the length of the shadow as the height. The angle between the hypotenuse and the base is your Sun angle.
  4. Now, you need to calculate the distance between your location and the location at which the sun is exactly overhead. This can be done by using the Ruler feature on Google Earth.

*Please note that it is highly recommended to carry out this experiment on a solstice or equinox in order to get accurate results. For example, if you perform the experiment on the summer solstice (21 June), you simply need the distance between your location and the Tropic of Cancer for step 4. Similarly, on the winter solstice (21 December) you need the distance between your location and the Tropic of Capricorn and on equinoxes (21 March and 23 September) you need your distance from the equator. However, if you won’t be performing the experiment on any of these days, then you need to check at which location the Sun will be exactly overhead on the particular date you are working on, and then calculate your distance from that location.

Now, simply plug in the appropriate values in the given formulae, and voilà! You have calculated the circumference of the Earth!

Editor’s note: If you are interested in doing the Eratosthenes experiment collaboratively on a large scale, check out the yearly experiment (schools are invited to register) hosted by the International Astronomical Union!

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