Eratosthenes Measurement with an Overstated Precision
At a district science fair, two students present an Eratosthenes-style measurement. School A is at 40.0°N, 88.0°W; School B is at 32.2°N, 88.0°W. On the equinox at local noon they measure the Sun's angle from vertical as 5.0° at A and 12.8° at B, each with an estimated reading uncertainty of ±0.4°. The road distance between the schools is 890 km; the great-circle distance is 870 km. Using 890 km and a 7.8° angle difference, the students calculate Earth's circumference as 41,100 km and report a precision of ±1%. They timed local noon by clock rather than by shortest shadow, and they sighted the Sun's upper limb rather than its center. A judge notes that the schools' longitudes differ by 0.2° and that the Sun's angular diameter is about 0.5°.
1. Evaluate whether the students' reported ±1% precision is supported by their measurements. Show the error propagation and identify the dominant source of uncertainty.
Read the worked answer
The central calculation is correct in form:
[C=\frac{360^\circ}{\Delta\theta}d=\frac{360}{7.8}(890\ \text{km})\approx 41{,}100\ \text{km}.]
A ±1% claim means ±410 km, which would require the angle difference to be known to about 0.08°. That is far smaller than the stated ±0.4° per reading. Even if the two angle errors are independent, the difference has an uncertainty of about (\sqrt{0.4^2+0.4^2}=0.57^\circ); a conservative linear sum is 0.8°. Since (\Delta C/C\approx \Delta(\Delta\theta)/\Delta\theta), the uncertainty is roughly 0.57/7.8 ≈ 7% to 0.8/7.8 ≈ 10%, or ±3,000 to ±4,100 km. Systematic effects add more: using 890 km instead of 870 km raises the answer by 2.3%; limb-versus-center sighting can shift each angle by up to about 0.25°, and clock timing changes the hour angle. The dominant uncertainty is therefore the angular measurement, not the distance. A defensible statement is about 41,000 km with roughly 10% uncertainty, not ±1%.
2. The students want to improve the experiment. Rank these changes by their likely effect on accuracy and justify your ranking: (a) use the great-circle distance, (b) measure the Sun's center with a pinhole projector, (c) repeat the measurement on many days and average, (d) choose sites on the same meridian.
Read the worked answer
Rank by addressing the dominant error first. (b) and (c) target the angular uncertainty, which is the largest term (roughly 7–10%); a pinhole projector removes the limb-versus-center ambiguity and repeated measurements reduce random pointing error. (d) also matters because a longitude offset makes the observed angle difference depend on the hour angle rather than latitude alone; 0.2° is small here but could matter at the 1% level. (a) removes a 2.3% systematic error and is easy, but it is smaller than the angle error. A reasonable ranking is b ≈ c > d > a. Averaging many days reduces random error by (1/\sqrt{N}), but it does not remove a consistent bias such as sighting the limb. The best plan combines all four: same meridian, great-circle distance, solar projection, and repeated measurements with a stated uncertainty budget.