Experiments

Measuring the circumference of the Earth

Measuring the circumference of the Earth is a classic experiment inspired by the work of Eratosthenes in ancient Greece. Here’s how you can perform a modern version of his experiment:

Materials Needed:

1. Two vertical sticks or poles of equal length.

2. A measuring tape or ruler.

3. A stopwatch or clock.

4. A GPS device or a reliable method to determine the exact distance between the two locations.

5. Access to two locations aligned roughly north-south (as far apart as feasible) and where sunlight can cast shadows.

Steps:

1. Select Two Locations:

• Choose two locations at a known distance apart along roughly the same meridian (longitude). For simplicity, they should be relatively far apart but in the same time zone.

• Examples could be two nearby cities or locations a few hundred kilometers apart.

2. Measure Shadow Lengths at the Same Time:

• At solar noon (when the sun is at its highest point in the sky), measure the length of the shadow cast by each stick in both locations. Solar noon can be determined using an online solar calculator.

• Record the length of the shadow and the height of the stick at each location.

3. Calculate the Angles:

• Use trigonometry to calculate the angle of the sun’s rays at each location: Angle=arctan⁡(Length of ShadowHeight of Stick)\text{Angle} = \arctan\left(\frac{\text{Length of Shadow}}{\text{Height of Stick}}\right)Angle=arctan(Height of StickLength of Shadow​)

• Calculate the difference between the two angles. This difference corresponds to the angular separation between the two locations on the Earth’s surface.

4. Measure the Distance:

• Use GPS or a reliable map to measure the straight-line distance (DDD) between the two locations.

5. Calculate the Earth's Circumference:

• The Earth is a sphere, so the proportion between the measured angle (θ\thetaθ) and the full circle (360°) equals the proportion of the measured distance to the Earth's circumference: Circumference of Earth=D×360∘θ\text{Circumference of Earth} = \frac{D \times 360^\circ}{\theta}Circumference of Earth=θD×360∘​

• Ensure the angle is in degrees and the distance is in consistent units (e.g., kilometers).

Example Calculation:

• Shadow lengths: Stick in Location A casts a shadow of 1.0 m with a 2.0 m stick (tan⁡−1(0.5)=26.57∘\tan^{-1}(0.5) = 26.57^\circtan−1(0.5)=26.57∘). Stick in Location B casts a shadow of 0.5 m with a 2.0 m stick (tan⁡−1(0.25)=14.04∘\tan^{-1}(0.25) = 14.04^\circtan−1(0.25)=14.04∘).

• Angular difference: 26.57∘−14.04∘=12.53∘26.57^\circ - 14.04^\circ = 12.53^\circ26.57∘−14.04∘=12.53∘.

• Distance between locations: 800 km.

• Circumference: Circumference=800×36012.53≈23,000 km.\text{Circumference} = \frac{800 \times 360}{12.53} \approx 23,000 \, \text{km}.Circumference=12.53800×360​≈23,000km.

Tips for Accuracy:

• Perform the experiment on the equinox to minimize errors from the Earth's axial tilt.

• Use precise measurements for shadow lengths and location distances.

• Choose a clear, sunny day with minimal atmospheric distortion.

This method is simple yet remarkably accurate, given the tools available!