Eratosthenes Earth Circumference Calculator

An interactive tool to reproduce the brilliant 240 BC calculation of Earth’s size using ancient geometry.

Visualizing the Angle

This chart illustrates the shadow angle as a slice of the Earth’s full 360° circle.

What is the Eratosthenes Earth Circumference Calculator?

The Eratosthenes Earth Circumference Calculator is a tool designed to replicate the historical method used by the ancient Greek scholar Eratosthenes of Cyrene around 240 BC to first measure the size of our planet. This was a monumental achievement in the history of science and geography, proving not only that the Earth was a sphere but also providing a remarkably accurate estimate of its circumference using simple geometry. This calculator is for students, educators, and history enthusiasts who want to understand the logic behind the how did Eratosthenes measure the Earth experiment.

Eratosthenes observed that on the summer solstice at noon, in the city of Syene (modern Aswan, Egypt), the sun’s rays shone directly down a deep well, casting no shadow. However, in Alexandria, located almost directly north of Syene, a vertical stick (a gnomon) cast a shadow at the same time. By measuring the angle of this shadow and knowing the distance between the two cities, he could calculate the planet’s total circumference.

The Formula for Calculating Earth’s Circumference

Eratosthenes’ genius was in recognizing that the difference in the sun’s angle between two locations on a sphere was directly proportional to the distance between them. He made two critical assumptions: the Earth is a perfect sphere, and the sun’s rays are parallel when they reach Earth. The formula is elegantly simple:

Circumference = (360° / α) × d

This formula, a cornerstone of the ancient Greek astronomy, established a new era of measurement.

Variables in the Eratosthenes Calculation

Variable	Meaning	Unit (Auto-Inferred)	Typical Range
α (alpha)	The shadow angle measured in the northern city (Alexandria).	Degrees (°)	1° – 15°
d	The north-south distance between the two cities.	stadia, km, miles	4000-6000 stadia
360°	The total number of degrees in a full circle.	Degrees (°)	Constant

Practical Examples

Example 1: Eratosthenes’ Original Calculation

Let’s use the numbers that Eratosthenes himself is believed to have used.

• Inputs:
  • Shadow Angle (α): 7.2°
  • Distance (d): 5,000 stadia
  • Units: stadia
• Calculation:
  • (360° / 7.2°) × 5,000 stadia = 50 × 5,000 stadia
• Result: 250,000 stadia. Many historians believe he adjusted this to 252,000 stadia to make it divisible by 360.

Example 2: A Modern Hypothetical Measurement

Imagine two modern cities on the same line of longitude, and we measure the distance in kilometers.

• Inputs:
  • Shadow Angle (α): 8°
  • Distance (d): 890 km
  • Units: kilometers
• Calculation:
  • (360° / 8°) × 890 km = 45 × 890 km
• Result: 40,050 km. This is incredibly close to the actual polar circumference of Earth (approx. 40,008 km). This demonstrates the power of the Earth circumference formula.

How to Use This Eratosthenes Earth Circumference Calculator

Using this calculator is simple. Follow these steps to perform your own calculation:

1. Enter the Shadow Angle: Input the angle you measured or want to test into the “Shadow Angle (α)” field. This must be in degrees.
2. Enter the Distance: Input the distance between your two measurement points in the “Distance between Cities (d)” field.
3. Select the Correct Units: Use the dropdown menu to choose whether your distance is in stadia, kilometers, or miles. The output will automatically match this unit.
4. Interpret the Results: The calculator instantly updates the “Estimated Earth Circumference” in the results box. It also shows intermediate values like the calculation multiplier (360/α) to help you understand the math.
5. Reset (Optional): Click the “Reset” button to return to Eratosthenes’ original values of 7.2° and 5,000 stadia.

Key Factors That Affect the Calculation

While the method is brilliant, its accuracy depends on several factors. Understanding these is key to appreciating the result of this Eratosthenes Earth Circumference Calculator.

• Accurate Angle Measurement: Even a small error in measuring the 7.2-degree angle can lead to a large error in the final circumference. The tools used, like the gnomon or sundial, had to be precise.
• Accurate Distance Measurement: The distance between Syene and Alexandria (5,000 stadia) was a known trade route distance, possibly measured by trained walkers called bematists. Any inaccuracy here directly impacts the result. Check out our stadia to km conversion tool for more context.
• Cities on the Same Meridian: The formula works best if the two cities are perfectly north-south of each other. Alexandria is actually about 3° west of Syene, which introduced a small error.
• Parallel Sun Rays: This assumption is very accurate because the Sun is so far away. The rays arriving at Earth are virtually parallel.
• Earth is a Perfect Sphere: The Earth is actually an oblate spheroid, slightly wider at the equator. For this calculation, however, assuming a perfect sphere is a very reasonable simplification that yields a close result.
• Timing of Measurement: The measurement must be taken at the exact same time (local noon) in both cities on the same day (the summer solstice). While they lacked instant communication, ancient astronomers were excellent at tracking time by the sun.

Frequently Asked Questions (FAQ)

1. How accurate was Eratosthenes’ original calculation?

Amazingly accurate. His result of 252,000 stadia translates to somewhere between 39,690 km and 46,620 km, depending on the exact length of the stadion he used. The actual circumference is about 40,075 km. His estimate was off by anywhere from 2% to 15%, an incredible feat for 240 BC.

2. What is a “stadion”?

The stadion (plural: stadia) was an ancient Greek unit of length, based on the length of a sports stadium. Its exact length varied by region, but it’s typically estimated to be between 157 and 185 meters. This uncertainty is the main reason for the range in his accuracy.

3. Why was the measurement taken on the summer solstice?

Eratosthenes chose the summer solstice because he knew Syene was located on or very near the Tropic of Cancer. This is the latitude where the Sun is directly overhead at noon on the solstice, meaning vertical objects cast no shadow. This created a perfect “zero point” for his experiment.

4. Can I replicate this experiment myself?

Yes! You can coordinate with a friend in a city several hundred miles directly north or south of you. On the same day at local noon, both of you measure the length of a shadow cast by a vertical stick of the same height. You can then calculate the sun’s angle and, using the distance between your cities, use this very Eratosthenes Earth Circumference Calculator to estimate the Earth’s size.

5. Does the height of the stick matter?

No, the height of the stick itself doesn’t directly enter the final circumference formula. You use the height of the stick and the length of the shadow to calculate the *angle* (specifically, angle = arctan(shadow_length / stick_height)). It is this angle that is crucial. A taller stick will produce a longer shadow, but the angle will be the same.

6. Why does the calculator default to 7.2 degrees?

7.2 degrees is the angle most commonly cited in historical texts for Eratosthenes’ measurement in Alexandria. It’s a convenient number because 7.2 is exactly 1/50th of a full 360-degree circle, simplifying the calculation.

7. What is the “Multiplier” in the results?

The multiplier is the result of the first part of the equation: (360° / Shadow Angle). It tells you how many times the distance between your two cities would fit into the full circumference of the Earth. For Eratosthenes’ 7.2° angle, the multiplier is 50.

8. What if the cities are not on the same meridian?

If the cities are not on the same line of longitude, it introduces a small error. The formula requires the true north-south distance. Since Alexandria was slightly west of Syene, the actual ground distance Eratosthenes used was slightly longer than the pure north-south distance, which would have slightly inflated his result.