Eratosthenes’ Earth Circumference Calculator

Discover the ancient method used to measure our planet.

Simulate The Experiment

What Did Eratosthenes Use to Calculate Earth’s Circumference?

More than 2,200 years ago, a brilliant Greek polymath and the chief librarian at the Library of Alexandria named Eratosthenes of Cyrene performed an experiment that answered a monumental question: what is the circumference of the Earth? Eratosthenes didn’t use satellites or modern GPS. Instead, what he used was a combination of simple observation, basic geometry, and a few key tools: a stick (or ‘gnomon’), his own intellect, and information about two cities in Egypt. His method was so elegant and his result so accurate that it remains one of the greatest intellectual achievements in history.

The Eratosthenes Formula and Explanation

The logic behind the calculation is based on a few core assumptions: that the Earth is a sphere, and that the Sun’s rays arrive at Earth in parallel lines. Knowing this, Eratosthenes devised a simple formula:

Earth’s Circumference = (Distance between Cities) × (360° / Shadow Angle)

He knew that on the summer solstice, the sun was directly overhead in the city of Syene (modern Aswan), as it illuminated the bottom of a deep well. At the exact same time in Alexandria, to the north, a vertical stick cast a measurable shadow. The angle of this shadow, he reasoned, must be the same as the angle separating the two cities at the Earth’s center.

Variables Used in the Calculation

The key inputs for Eratosthenes’ calculation.

Variable	Meaning	Unit	Eratosthenes’ Value
Shadow Angle	The angle cast by a vertical object at the northern location (Alexandria).	Degrees	~7.2° (or 1/50th of a circle)
Distance	The north-south distance between the two observation points (Syene and Alexandria).	Stadia	~5,000 stadia

Practical Examples

Example 1: Eratosthenes’ Original Calculation

Using the numbers historians believe Eratosthenes used:

• Inputs: A shadow angle of 7.2 degrees and a distance of 5,000 Greek stadia.
• Calculation: (5,000 stadia) * (360 / 7.2) = 250,000 stadia.
• Result: If we use the Greek stadion (185m), this translates to 46,250 km, which is about 15% larger than the actual circumference. If he used the Egyptian stadion (157.5m), his result is 39,375 km—an error of less than 2%!

Example 2: A Modern Hypothetical

Imagine two modern cities directly north-south of each other, 900 km apart.

• Inputs: On a given day, you measure a sun shadow angle of 8.1 degrees in the northern city. The distance is 900 km.
• Calculation: (900 km) * (360 / 8.1) = 40,000 km.
• Result: This calculation yields an incredibly accurate circumference of 40,000 km, very close to the true value of 40,075 km.

How to Use This Eratosthenes Calculator

1. Enter Shadow Angle: Input the angle of the sun’s shadow as measured in the northern location. Eratosthenes used 7.2 degrees.
2. Enter City Distance: Input the known distance between the two north-south locations.
3. Select Units: Choose the unit of measurement for your distance. This is critical, as the “stadion” had different lengths. See how switching from the Greek to Egyptian stadion makes the result much more accurate.
4. Interpret Results: The calculator automatically shows the Earth’s circumference in modern units and compares it to the accepted value, showing the percentage of error.

Key Factors That Affect the Calculation

Eratosthenes’ method was brilliant, but several factors could introduce errors:

• Distance Accuracy: The distance of 5,000 stadia was an estimate, likely from professional walkers (bematists) who were trained to walk in equal steps. Any error here multiplies through the whole calculation.
• Angle Measurement: Measuring the 7.2-degree angle with the tools of the time (likely a gnomon or scaphion) was challenging. A small error in the angle leads to a large error in the circumference.
• City Alignment: Alexandria is not perfectly north of Syene. This east-west deviation introduces a small error.
• Timing of Measurement: The measurements needed to be taken at local noon on the same day.
• Earth’s Shape: The Earth is not a perfect sphere; it is an oblate spheroid (slightly flattened at the poles). This was a minor source of error.
• The Sun’s Rays: The assumption that the sun’s rays are parallel is extremely accurate and not a significant source of error due to the vast distance of the Sun.

Frequently Asked Questions (FAQ)

How did Eratosthenes measure the distance between cities?

He likely relied on data from bematists, surveyors trained to walk with uniform steps to measure long distances.

What tools did he use to measure the angle?

He would have used a gnomon (a vertical stick used as a sundial) or possibly a scaphion (a sundial bowl). The angle was derived from the length of the shadow relative to the height of the stick.

Why did he choose Syene and Alexandria?

Syene was famous because it lay on the Tropic of Cancer, meaning the sun was directly overhead at noon on the summer solstice. Alexandria was a major center of learning and located a significant, measurable distance to the north.

How accurate was Eratosthenes’ calculation?

Amazingly accurate for his time. Depending on which ‘stadion’ length he used, his estimate was off by anywhere from 1% to 16%.

Can I replicate this experiment today?

Yes! With a partner in a city several hundred miles north or south of you, you can perform the same measurement on the same day and get a surprisingly good result.

Did Eratosthenes know the Earth was round?

Yes, most Greek scholars of the era knew the Earth was a sphere. They observed the curved shadow of the Earth on the moon during a lunar eclipse and saw ships disappear over the horizon hull-first.

Does the unit of the ‘stadion’ really matter?

It matters immensely. As our calculator shows, switching between the ~185m Greek stadion and the ~157.5m Egyptian stadion changes the result’s accuracy from ~15% error to less than 2% error.

What is the main principle behind the formula?

The principle is that the ratio of the distance between the two cities to the entire circumference of the Earth is equal to the ratio of the measured shadow angle to the 360 degrees of a full circle.

Related Tools and Internal Resources

• Prime Number Sieve: Explore the Sieve of Eratosthenes, another invention by the same scholar.
• Axial Tilt Calculator: Learn about the tilt of the Earth, which Eratosthenes also calculated with remarkable accuracy.
• History of Geodesy: A deep dive into the science of measuring the Earth.
• Ancient Greek Astronomy: Discover the astronomical knowledge of the ancient world.
• Tools of Ancient Science: Learn about the instruments used by early scientists.
• The Library of Alexandria: Read about the center of knowledge where Eratosthenes worked.