# Eratosthenes’ shadow

Some days the world feels really big and some days it feels quite small, but as you might suspect, it doesn’t actually change size. Here in the 21st century, we know the circumference, density, and other physical properties with great accuracy. However, back in 3 BC even the first of these was still a mystery. That is, until a man named Eratosthenes devised a simple yet elegant way to calculate the size of the Earth using nothing but the sun and some mad geometry skills.

My friend Gene Gordon, a high school physics teacher in upstate New York, got me excited about the simplicity of Eratosthenes’ original experiment. Gene had the great idea to use social media to get students all over the country (and even the world) to recreate Eratosthenes’ experiment “simultaneously.”

I’m going to let the one and only Carl Sagan give you the full background on Eratosthenes farther down the post, but in the spirit of Gene’s lesson plan, I want to demonstrate how you, too, can determine the size of the Earth with a little help from your friends.

Now comes the math – it’s after the jump. I promise it won’t hurt.

1) Calculate the angle the sun’s rays hit your stick at solar noon using the equation given below. You and your friend should each do this separately. This will give you TWO ANGLES, one for the person at the higher latitude location and one for the person at the lower latitude location. [Hint: the function below is called “arctan” and appears on any scientific calculator.]

2) Whichever one of you has the smaller angle, subtract this from the bigger one. This is your THETA.

3) Find the distance in miles or kilometers between the LATITUDE of your location and the LATITUDE of your friend’s location. (DO NOT find the distance as the crow flies.) This is your DISTANCE.

4) Now for the grand finale! Your THETA represents a portion of a full circle, a fraction of 360 degrees. Your DISTANCE is the equivalent fraction of the Earth’s circumference. So divide 360 by your THETA and multiply by your DISTANCE. TA-DA! You’ve calculated the circumference of the Earth. [Note: your circumference will be in the same units as your stick and shadow so a little unit conversion may be in order.]

Eratosthenes actually had it easy, he already knew of a nearby city (Syene) where the sun was directly overhead at midday on one day of the year - the summer solstice - meaning that all he had to do was measure the angle of the sun in Alexandria, where he was.

Because the sun is so far away from Earth, incoming light rays are essentially parallel. And thanks to geometry, we have a handy-dandy theorem for determining angles when two parallel lines are transected by a third line: alternate interior angles are congruent (or equal). In practical terms, this means that the angle measured by Eratosthenes in Alexandria is equal to the angular distance between Alexandria and Syene. To get the last piece of the puzzle, Eratosthenes paid a guy to actually pace out the distance between the two cities. Et voila! He had all he needed to calculate the circumference of the Earth. And he was surprisingly accurate, coming well within 10% of the present day value of ~40,000 km (depending on historical interpretations).

But enough math for now! I’ll leave you with how Carl tells the story of Eratosthenes, which is of course beautifully.

*Summer Ash is The Rachel Maddow Show’s in-house astrophysicist. She blogs at Newtonianism for the Ladies.*