Eratosthenes’ Measurement of the Earth’s Circumference

Introduction

Around 240 BCE, Eratosthenes of Cyrene pioneered one of the earliest measurements of our planet’s size. By combining solar angle observations at two locations and knowing the distance between them, he estimated the Earth’s circumference with remarkable accuracy.

Historical Timeline

• 3rd century BCE: Eratosthenes serves as chief librarian at the Great Library of Alexandria.
• Circa 240 BCE: Observes summer solstice sun angles in Syene and Alexandria.
• Measurement: Records angle α ≈ 7.2° and distance D ≈ 5,000 stadia.
• Result: Calculates Earth’s circumference ≈ 250,000 stadia (~39,375 km).

Original Measurements

On the summer solstice, the Sun was directly overhead at Syene (now Aswan), illuminating the bottom of deep wells. In Alexandria, vertical sticks cast shadows, indicating an angular deviation α.

• Angle α: ~7.2° (1/50 of 360°).
• Distance D: ~5,000 Egyptian stadia (~787 km).

How the Math Worked

The central angle α corresponds to the fraction α/360 of the full circumference C:

C = 360° × D / α

Plugging in D=5,000 stadia and α=7.2°:

C ≈ 360° × 5,000 stadia / 7.2° ≈ 250,000 stadia ≈ 39,375 km

Online Calculator Tool

Enter your measured angle and distance to compute Earth’s circumference in stadia, kilometers, and miles.

Summary

Eratosthenes’ ingenious use of shadows and simple geometry provided the first scientific estimate of Earth’s size. His method remains a classic demonstration of observational science.

Evidence of Modern-Day Usage

• Educational experiments replicate Eratosthenes’ method in schools worldwide.
• Citizen science projects coordinate measurements across continents to refine Earth models.
• Mathematics courses use his calculation as a classic applied trigonometry example.