By Karren Doll Tolliver • Feb 15, 2023 4:30 pm EST
Image credit: JavierHuras/iStock/GettyImages
Using the curvature of Earth and the Sun’s parallel rays, you can measure shadows at two points to calculate the planet’s radius with minimal equipment.
In 240 BCE, Greek mathematician Eratosthenes famously estimated Earth’s circumference by comparing shadow angles in Syene (modern‑day Aswan) and Alexandria. By knowing the distance between the two locations and the angle difference, he derived a circumference of approximately 39,350 km and a radius of about 6,267 km. Today, anyone with a simple pole and a protractor can replicate this historic experiment.
Measure the straight‑line distance (arc length) between your site and a partner’s site that lies roughly along the same meridian. In Eratosthenes’ original experiment, the distance between Syene and Alexandria was 787 km. Use any consistent unit of measurement; the proportional relationship remains unchanged.
Drive each pole into the ground so that it stands perfectly vertical. Attach a string to the top of each pole. The free end of the string will be used to trace the tip of the shadow cast by the pole.
Because the Sun’s position changes with time, both observers must record their measurements at the exact same moment. If you are in different time zones, adjust the local time accordingly (e.g., a 2‑hour difference requires a 2‑hour offset). It is safest to use a shared digital clock or an online time‑sync service.
At local solar noon—when the Sun is highest in the sky and shadows are shortest—place the free end of the string at the tip of the shadow and tighten it. Use the protractor to read the angle between the pole and the string at the top. Record the angle in degrees. Your partner should perform the identical procedure at the same instant.
Subtract the two recorded angles to find the angular difference (Δθ). In Eratosthenes’ case, Δθ was 7.2°.
Since the two points lie on a circle around Earth, the arc length (distance measured) corresponds to Δθ degrees out of a full 360° circle. Set up the proportion:
\(\frac{Δθ}{360°} = \frac{distance}{C}\)
Solving for C (circumference):
C = \(\frac{distance \times 360°}{Δθ}\)
With distance = 787 km and Δθ = 7.2°, the calculation yields a circumference of approximately 39,350 km.
Use the relationship between circumference and radius:
C = 2πr
Rearrange to r = C / (2π). Plugging in C = 39,350 km gives:
r ≈ 6,267 km.
While this method is historically significant, it introduces several practical errors:
Modern geodesy shows Earth’s equatorial radius to be 6,378.1 km and the polar radius 6,356.7 km, reflecting its slightly flattened shape. Satellite altimetry and GPS provide far more precise measurements.
Today, scientists employ satellite gravimetry, laser ranging, and global positioning systems to determine Earth’s dimensions with millimeter accuracy. Nonetheless, the shadow‑measurement experiment remains a valuable educational demonstration of scientific methodology.
Recreating Eratosthenes’ experiment connects you with a centuries‑old scientific legacy and illustrates the power of simple observations to unlock planetary truths. While the resulting numbers will be approximate, the process offers insight into geometry, astronomy, and the history of measurement.
