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 Earth's size.
But the famous phrase “he measured Earth with a shadow” hides the most interesting part.
The shadow was only the local clue. Geometry turned that clue into a global measurement.
Quick Answer
Eratosthenes compared the Sun's direction at two Egyptian locations, Alexandria and Syene.
In the surviving ancient account, the angular difference was treated as 1/50 of a full turn, while the distance between the locations was taken as 5,000 stadia.
If 5,000 stadia represents 1/50 of Earth's circumference, then:
Earth's circumference ≈ 50 × 5,000 = 250,000 stadia
The shadow did not measure Earth by itself. A local angle, a baseline distance, and a geometric model worked together.
The Structural Question — How Can a Local Measurement Reveal a Global Size?
This article is really about a powerful scientific pattern:
local observation + known baseline + model → hidden global quantity
Eratosthenes could not walk around the entire planet with a measuring rope.
He measured something reachable, connected it to a model of Earth, and inferred something unreachable.
This pattern appears again in surveying, astronomy, navigation, remote sensing, and engineering.
The Question Was Not “Is Earth Round?”
Greek thinkers had discussed a spherical Earth before Eratosthenes.
His measurement used a spherical Earth as part of the model.
The problem was:
How big is the sphere?
Eratosthenes worked in Alexandria in the third century BCE. His own account of the measurement is lost. The most detailed surviving account is later, especially through Cleomedes.
That historical gap matters. We should distinguish the elegant method preserved in later sources from details that popular retellings sometimes add with more confidence than the evidence allows.
Two Places, Two Local Verticals
The simplified surviving story uses Alexandria in the north and Syene, near modern Aswan, farther south.
At Syene, around noon near the summer solstice, the Sun was treated as essentially overhead.
At Alexandria, the Sun was not directly overhead, so a vertical gnomon produced an angular offset.
The surviving geometric account treats that offset as:
1/50 of a full circle
In modern degree language:
360° / 50 = 7.2°
But this is a modern translation. It is safer not to imagine Eratosthenes literally writing “7.2°” in modern notation.
Why Does the Shadow Angle Equal an Angle Inside Earth?
This is the step that many readers want to see, not just memorize.
First, treat sunlight arriving at the two cities as nearly parallel because the Sun is extremely far away compared with the distance between the cities.
Second, a vertical direction at each city points along a radius of the spherical Earth.
At Syene, the sunlight is treated as aligned with the local vertical.
At Alexandria, the sunlight differs from the local vertical by the observed angle.
Because the sunlight directions are parallel, that same angular difference appears between the two local verticals—and therefore between the two radii drawn from Earth's center.
So a small shadow angle at the surface becomes a central angle of the planet.
Figure 1. Modern reconstruction, not to scale. The local angular difference is interpreted as the same fraction of Earth's full turn.
Could One Shadow Alone Measure Earth?
No.
This is an important correction to the popular shorthand.
One shadow can tell you the Sun's direction relative to your local vertical.
To infer Earth's circumference, you also need:
- a second location or an equivalent reference condition;
- the surface distance between the locations;
- a model connecting local vertical directions to Earth's center;
- the assumption that the Sun's rays are effectively parallel.
The measurement is powerful precisely because several modest pieces of information combine.
The Whole Calculation Fits in One Proportion
If the angular separation is 1/50 of a full turn, then the north-south surface arc between the cities is modeled as 1/50 of Earth's circumference.
angle / full turn = arc distance / circumference
So:
1 / 50 = 5,000 stadia / C
and therefore:
C = 250,000 stadia
Cleomedes reports 250,000 stadia. Other ancient sources associate a value of 252,000 stadia with Eratosthenes.
Where Did the 5,000 Stadia Come From?
This is another point where popular retellings often sound more certain than the surviving evidence.
The ancient calculation uses a distance of 5,000 stadia between Alexandria and Syene.
But Eratosthenes' own account is lost, and historians debate exactly how that baseline was obtained and which information he measured himself versus inherited from geographic or surveying knowledge.
So the safest statement is not “we know exactly who paced the route.”
It is:
the method requires a baseline distance, and the surviving account uses 5,000 stadia
Why We Should Be Careful with Kilometers
It is tempting to convert 250,000 stadia directly into modern kilometers and announce a precise percentage error.
The problem is that the exact length of the stade used in this context is uncertain.
Different ancient stade values produce different modern conversions.
That makes the ancient result:
about 250,000 stadia
historically more secure than one exact modern kilometer conversion.
The central achievement is the method, not a modernized accuracy headline.
The Assumptions Matter
This is where the previous articles on proof become useful.
The inference works because a set of assumptions links the observation to the conclusion:
- Earth is approximately spherical.
- Sunlight reaching the two locations is effectively parallel.
- Alexandria and Syene can be treated as lying on the same north-south meridian.
- The distance between the locations is known well enough.
- The angular difference is known well enough.
Some are approximations, not exact statements about the real world.
Syene is not exactly on the Tropic of Cancer. Alexandria and Syene are not exactly on the same meridian. The baseline and angle also have measurement uncertainty.
A model can be useful without being perfectly exact—but we should know which simplifications connect the measurement to the answer.
Which Error Matters More?
The modern form of the model is:
C = d × 360° / θ
where d is the baseline distance and θ is the angular difference.
This equation immediately tells us how uncertainty behaves.
If the distance is 1% too large, the circumference estimate is 1% too large.
If the angle is too large, the circumference estimate becomes too small because θ is in the denominator.
For example, with d = 800 km:
| Angle | Estimated circumference |
|---|---|
| 7.1° | ≈ 40,563 km |
| 7.2° | 40,000 km |
| 7.3° | ≈ 39,452 km |
A change of only 0.1° around 7.2° shifts the estimate by roughly 1.4%.
This is a first glimpse of sensitivity: how strongly an inferred answer responds to uncertainty in an input.
Try the Geometry with Modern Units
Forget ancient units for a moment.
Suppose two locations are 800 km apart along the relevant north-south arc, and their Sun-direction difference is 7.2°.
Then:
C ≈ 800 × 360 / 7.2 = 40,000 km
This is a modern numerical illustration of the geometry.
It is not a claim that Eratosthenes' 250,000 stadia can be converted uniquely into exactly 40,000 km.
Python — Change the Inputs
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 before you run it.
Keep the distance at 800 km. Change the angle from 7.2 to 8.0.
Will the estimated Earth become larger or smaller?
Then keep 7.2° fixed and increase the distance by 1%. Which change affects the estimate in a simpler way?
Could We Repeat the Idea Today?
Yes.
A modern classroom version can compare Sun angles measured from two separated locations and combine the angular difference with the distance between them.
The details matter: vertical sticks, local solar timing, location geometry, and distance all affect accuracy.
That is part of the lesson rather than a nuisance.
The experiment teaches not only geometry, but measurement design.
From a Shadow to Modern Geodesy
The modern field that measures Earth's size, shape, orientation, gravity field, and positions is called geodesy.
Modern geodesists use satellites, GPS, precise clocks, lasers, gravity measurements, and global reference systems.
The instruments are radically different from Eratosthenes' world.
But the underlying questions are recognizable:
Where are we?
How far apart are two places?
What shape are we measuring?
How can local observations constrain a global model?
Why This Story Matters
The famous shadow is memorable, but the deeper lesson is the inference architecture.
Eratosthenes could not directly measure Earth's full circumference.
He could access a local angle and a baseline.
The spherical model connected them to the hidden global quantity.
Measure what you can reach. State the model. Use the model to infer what you cannot reach directly.
Words to Keep
gnomon
A vertical object or instrument used to produce a measurable solar shadow.
baseline
A known distance used together with another measurement to infer an unknown quantity.
central angle
An angle whose vertex is at the center of a circle or sphere.
proportion
An equality between two ratios.
geodesy
The science of accurately measuring and understanding Earth's shape, orientation, gravity field, and positions.
sensitivity
How strongly an output changes when an input measurement changes.
model
A simplified mathematical representation used to connect observations to quantities we want to infer.
One Sentence to Keep
Eratosthenes combined a local angular difference, a baseline distance, and a spherical Earth model to scale a reachable measurement into a planetary circumference.
What Should We Ask Next?
Eratosthenes used a shadow to infer a distance far larger than anything he could measure directly.
But the same idea can be brought back down to everyday scale.
Can a shadow tell us the height of a tower or another object we cannot directly reach?
Next: 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” — Eratosthenes' historical context, lost original work, later accounts, and the 250,000-stadia result.
- Mathematical Association of America, “Eratosthenes and the Mystery of the Stades” — assumptions, historical uncertainties, the stade problem, and cautions about popular reconstructions.
- MAA, “Angle between shadow and staff in Alexandria” — the 1/50-turn angular statement and historical caution about modern degree notation.
- NOAA National Geodetic Survey, “What is geodesy?” — modern geodesy, GPS, coordinate measurement, and Earth's shape and gravity field.
- NASA, “Using the Eratosthenes Method” — educational reconstruction of the shadow-measurement method.
Research note: Reddit, YouTube, and Mathematics Stack Exchange were used as question sensors for recurring reader confusion—especially “how can one shadow reveal Earth's size?”, “why are the rays parallel?”, “where did the distance come from?”, “was the answer really almost exact?”, and “can the experiment be repeated today?”. They were not used as factual authorities.
Historical note: Eratosthenes' original account does not survive. The 7.2° notation, kilometer examples, sensitivity table, Python code, and diagram are modern teaching tools.