Eratosthenes measured Earth by connecting a local shadow angle to a much larger arc of the planet. In the familiar reconstruction, the Sun was overhead at Syene while a vertical object in Alexandria cast a shadow corresponding to about 7.2 degrees. If that angle represented one-fiftieth of a full circle, the distance between the locations represented one-fiftieth of Earth’s circumference.
The geometry is straightforward. The exact historical measurement is less secure: Eratosthenes’ own account has not survived, later writers report different totals, and the length of the ancient distance unit is uncertain. A confident modern percentage for his accuracy can hide those uncertainties.
How a shadow becomes an angle at Earth’s center
Sunlight arriving at Earth can be treated as nearly parallel for this calculation. On a spherical Earth, vertical directions at two separated locations point away from different positions on the curved surface. A difference between their solar angles therefore reveals the angle between those verticals.
In the simplified Alexandria–Syene construction, one vertical aligns with the incoming sunlight. The other is tilted by the measured shadow angle. Extending the two verticals toward Earth’s center makes the connection: the same angle measures the fraction of a circle between the sites.
OpenStax’s account of ancient astronomy presents this proportional argument. It assumes a spherical Earth; it is not a demonstration that the two shadow measurements alone, without a model of the Sun’s distance, settle every possible model of Earth’s shape.
If the measured angle is θ and the corresponding surface distance is d, the relationship is:
Circumference = d × 360° / θ
The distance must match the arc represented by the angle. A travel route with bends is not automatically that arc.
Working through the familiar numbers
An angle of 7.2 degrees is 1/50 of 360 degrees. Multiplying a distance of 5,000 stadia by 50 gives 250,000 stadia. Ohio State’s reconstruction explains this version of the calculation and its transmission through the later writer Cleomedes.
Here is a separate classroom example using kilometers. These are deliberately chosen demonstration values, not a claim that we have recovered Eratosthenes’ original measurements:
| Assumed distance | Assumed angle | Calculated circumference |
|---|---|---|
| 800 km | 7.2° | 40,000 km |
| 760 km | 7.2° | 38,000 km |
| 800 km | 7.0° | About 41,143 km |
The first two rows show that a 5% reduction in distance produces a 5% reduction in the result. The last row shows why the angle matters too: a smaller angle makes the same distance represent a smaller fraction of the whole circle, so the inferred circumference grows.
Multiplication does not remove uncertainty from the inputs. It carries that uncertainty into the answer.
Why “accurate to within 1%” needs more evidence
Three different questions are often compressed into that claim.
Which reported total are we using? The familiar Cleomedes reconstruction yields 250,000 stadia. Pliny’s Natural History, Book II, section 247 instead attributes 252,000 stadia to Eratosthenes. Pliny is an ancient witness to that reported value, not a surviving notebook of the original experiment. Choosing one number without identifying its source erases a real difference in the record.
How long was a stadion? Ancient units did not have one universally secure conversion into modern meters. Ohio State discusses how different choices change the inferred circumference. A seemingly excellent agreement can depend heavily on the conversion selected; it is not an independent confirmation of that conversion.
How idealized is the surviving explanation? Treating Syene as exactly beneath the solstice Sun, both cities as lying on one meridian, and their separation as a precisely known north–south arc makes the diagram convenient. A real measurement has to examine each approximation. The later account does not justify inventing a personal journey, an exact surveying procedure or an error bar for Eratosthenes.
The historical uncertainty does not make the method worthless. It changes the strength of the claim we can make: the proportional reasoning is reproducible, while some details of the ancient execution remain uncertain.
What a modern shadow experiment would have to control
A useful reconstruction begins with two locations whose separation and latitude difference are known. Observe shadows cast by vertical objects on level surfaces; do not look at the Sun. For the simplest north–south model, compare the Sun’s angle at each site’s local solar noon. Local solar noon is not automatically 12:00 on a civil clock, and sites at different longitudes reach it at different times.
Then ask what distance your calculation needs. Is it a meridian arc, or have you inserted a road distance? Are the objects actually vertical? Are the shadow tips sharp enough to measure? Did both observations use the same date and a compatible geometry? Those checks turn a striking demonstration into a defensible measurement.
This distinction between a calculation and the evidence supporting it also appears in our discussion of prediction and explanation in classical astronomy. Eratosthenes’ lasting contribution is the scale connection: an angle measured beside a shadow can constrain the size of a planet, provided the model, distances and uncertainties travel with the answer.
AI-generated illustration for Galileo Whispers of Earth from space. Geography reference: NASA Johnson Space Center’s Apollo 17 Blue Marble. This is not a NASA image or a historical measurement.
