Today, satellites can look back at Earth. GPS can tell us where we are within seconds.
Now remove all of that.
No satellite. No GPS. No photograph of Earth from space.
More than 2,000 years ago, Eratosthenes still found a way to estimate the size of Earth.
He needed a shadow, a distance, and one beautiful piece of geometry.
The Question Was Not “Is Earth Round?”
Greek thinkers had discussed a spherical Earth long before Eratosthenes.
His problem was different:
How big is it?
Eratosthenes worked in Alexandria in the third century BCE. His own account of the measurement is lost. The most detailed surviving ancient description comes from the later writer Cleomedes.[1]
Eratosthenes could not see the whole Earth. He measured one small angle and used geometry to connect it to the whole planet.
Two Places, Two Shadows
The story uses two Egyptian locations: Alexandria in the north and Syene, near modern Aswan, farther south.
At Syene, around noon on the summer solstice, the Sun was observed to be almost directly overhead. A vertical object cast essentially no shadow.
At Alexandria, a vertical gnomon did cast a shadow.
That shadow showed that the Sun was away from the local vertical by about:
1/50 of a full circle
In modern degree language, that is:
360° / 50 = 7.2°
Cleomedes reports the distance between Alexandria and Syene as 5,000 stadia.[1]
Why Does a Shadow Tell Us an Angle Inside Earth?
The Sun is so far away that its rays reaching Earth can be treated as nearly parallel.
Now imagine extending the two vertical directions at Alexandria and Syene downward. On a spherical Earth, they point toward Earth's center.
The same angle that appears between the sunlight and the vertical at Alexandria therefore corresponds to the central angle between the two locations.
This is the key step.
Figure 1. Schematic, not to scale. The measured shadow angle represents the same fraction of Earth's full turn.
The Whole Calculation Fits in One Proportion
If the angle between the two places is 1/50 of a full circle, then the surface distance between the places should also be about 1/50 of Earth's circumference.
angle / full turn = distance / circumference
So:
1 / 50 = 5,000 stadia / circumference
Multiply the distance by 50:
circumference = 250,000 stadia
Cleomedes gives this value. Other ancient sources report a value of 252,000 stadia associated with Eratosthenes.[2]
Why We Should Be Careful with Kilometers
It is tempting to convert 250,000 stadia directly into modern kilometers.
The problem is that the exact length of the stade used in this calculation is uncertain. Different ancient regions used different stade lengths, and historians still discuss which value is appropriate.[3]
So the most secure historical statement is the ancient one:
about 250,000 stadia
The lasting achievement is the method.
The Assumptions Matter
This is where our two Euclid articles become useful.
Eratosthenes' reasoning depended on assumptions.
- Earth is approximately spherical.
- Sunlight reaching the two places is effectively parallel.
- Alexandria and Syene can be treated as lying on the same north-south meridian.
- The distance between the locations is known.
- The shadow angle is known.
Some of these assumptions are only approximate.
Syene is not exactly on the Tropic of Cancer. Alexandria and Syene are not exactly on the same meridian. The historical distance estimate also had uncertainty.[2]
Yet the geometric method itself is sound.
A good model does not require the world to be perfectly simple. It requires us to know what we simplified and how that may affect the answer.
Try the Method with Modern Units
Forget ancient units for a moment.
Suppose two locations on a nearly north-south line are about 800 km apart.
Suppose their noon shadow observations differ by:
7.2°
Since 7.2° is 1/50 of 360°:
Earth circumference ≈ 50 × 800 km = 40,000 km
This is a modern numerical illustration of the geometry, not a conversion of Eratosthenes' stade value.
Python — Change the Angle
The calculation can be written in one short function.
def earth_circumference(distance, angle_deg):
return distance * 360 / angle_deg
distance_km = 800
angle_deg = 7.2
result = earth_circumference(distance_km, angle_deg)
print(result, "km")Predict first.
Keep the distance at 800 km. Change the angle from 7.2 to 8.0.
Will the estimated Earth become larger or smaller? Run the code and explain why.
From a Shadow to Modern Geodesy
The modern field that measures Earth's size, shape, orientation, gravity field, and positions on its surface is called geodesy.[4]
Today, geodesists use technologies that Eratosthenes could not have imagined: satellite signals, precise clocks, lasers, remote sensing, and global coordinate systems.
GPS is one part of that modern world. NOAA notes that modern geodesists use space-based systems such as GPS to determine positions on Earth's surface, while geodetic reference systems support mapping, surveying, construction, transportation, and navigation.[4]
The instruments changed completely.
The deeper questions did not:
Where are we?
How far apart are two places?
What shape are we measuring?
How does a local measurement reveal something global?
Why This Story Matters
Eratosthenes did not travel around Earth with a measuring rope.
He measured something small enough to reach: a shadow.
Then geometry connected that local measurement to something far too large to measure directly.
This pattern appears again and again in science and engineering.
Measure what you can reach. Build a model. Use the model to infer what you cannot reach directly.
Words to Keep
gnomon
A vertical object used to create a measurable shadow.
circumference
The distance around a circle.
proportion
An equality between two ratios.
geodesy
The science of accurately measuring and understanding Earth and positions on its surface.
model
A simplified mathematical representation used to reason about a real system.
One Sentence to Keep
Eratosthenes measured a small shadow angle, treated it as a fraction of Earth's full turn, and scaled a known distance into the size of the planet.
What Should We Ask Next?
Eratosthenes used a shadow to measure something he could never reach directly.
But the trick is more general than Earth.
Can the same geometry measure the height of a tower, the width of a river, or the distance to something we cannot touch?
Our next question is:
How Can a Shadow Measure Something You Cannot Reach?
Previous: What Is a Mathematical Proof Really Doing?
Sources & Further Reading
- MacTutor History of Mathematics, “Eratosthenes” — historical overview and surviving-source context.
- D. R. Dicks, “Eratosthenes,” Dictionary of Scientific Biography reprint — detailed discussion of Cleomedes' account, the 1/50 angle, 5,000 stadia, and model approximations.
- Mathematical Association of America, “Eratosthenes and the Mystery of the Stades” — reconstructed assumptions and discussion of the stade problem.
- NOAA National Geodetic Survey, “What is geodesy?” — modern geodesy, GPS, mapping, surveying, transportation, and navigation.
- NASA, “Using the Eratosthenes Method” — educational reconstruction using shadow observations.
Historical note: Eratosthenes' original account does not survive. The diagram, modern degree notation, kilometer example, and Python code are modern teaching reconstructions.