Hexagon Lab 01: A Shadow of a Doubt

Over 2,200 years ago, Eratosthenes, working at the Library of Alexandria, made a groundbreaking measurement. He learned that in Syene, Egypt, the sun was directly overhead at noon on the summer solstice, while in Alexandria, a tall obelisk cast a shadow at the same time. Eratosthenes realized that the difference in shadow angles between the two cities was due to Earth's curvature. Using this observation and the known distance between the cities, he estimated the circumference of the Earth.

Course: HPHYS
Unit: 1
Purpose

To use geometric reasoning and proportional relationships to estimate the circumference of the Earth, following Eratosthenes's method.

Materials
  • Calculator
  • Pencil and paper
  • Research Log
Finding the Angle
  1. Calculate the difference in shadow angles between Alexandria and Syene
  2. Determine the fraction of Earth's circle this wedge represents
    Hint: Fraction = Angle Difference / 360°
Calculating Earth's Circumference
  1. Set up a proportion where the 800 km distance represents the same fraction of Earth's total circumference as the calculated angle
    $\frac{\text{Angle Difference}}{360^\circ} = \frac{800\,\text{km}}{\text{Total Circumference}}$
  2. Solve for Earth's circumference by rearranging the proportion
Accounting for Uncertainty
  1. Consider that Eratosthenes's measurement of distance was the most uncertain part
  2. Calculate Earth's circumference if the distance was 10% larger (880 km)
  3. Calculate Earth's circumference if the distance was 10% smaller (720 km)
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