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Investigating Science 11–12 · Year 12

Measuring the Earth with two shadows, the Eratosthenes method

Module 5: Scientific Investigations

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The idea

Two shadow angles measured at local noon at places a known north–south distance apart give the Earth’s circumference by geometry, the method Eratosthenes chose because the Earth could not be measured directly.

What you need

  • a 1.000 m stick set vertical with a plumb line and spirit level on level ground, chalk, 2 m tape measure
  • a partner school several hundred kilometres north or south (for example Sydney and Brisbane) doing the same on the same day
  • the Geoscience Australia geodetic calculator (or a map) for the north–south distance between the two sites

How to do it

  1. On an agreed day near an equinox, mark the tip of the shadow every 2 minutes for 40 minutes around local noon.
  2. Find the shortest shadow and measure it to the nearest millimetre.
  3. Calculate the Sun’s angle from the vertical, θ = arctan(shadow/stick length).
  4. Exchange results with the partner school and find the difference in angle Δθ.
  5. Find the north–south distance d between the two sites and calculate C = (360°/Δθ) × d.
  6. Estimate the uncertainty from the shadow reading and compare with the accepted value; evaluate why the method suited Eratosthenes’ question.

What you should see

At an equinox (declination 0°) a 1.000 m stick in Sydney (33.87° S) casts a noon shadow of 0.671 m and in Brisbane (27.47° S) 0.520 m, so Δθ = 6.40°. The north–south distance between those latitudes on the WGS84 ellipsoid is 709.5 km, giving C = 39 910 km, within 0.3 % of the 40 008 km meridian circumference. A 1 mm shadow error shifts the Sydney angle by 0.04°, about 0.6 % of Δθ. NASA’s account of Eratosthenes prints the shadow angle as 7.12 degrees and calls it about one fiftieth of a circle; one fiftieth of 360° is 7.2°, or 7°12′, and 360/7.12 is 50.6, so the printed figure is most likely 7°12′ written as a decimal.

What changes

What you change
latitude of the site
What you measure
shortest noon shadow and sun angle
What you keep the same
  • same stick length
  • vertical stick on level ground
  • same date
  • shortest shadow, not clock noon

Common misconceptions

Each of these ideas is wrong, and the activity is a chance to test it.

  • People in ancient times thought the Earth was flat.
  • Local noon is 12:00 on the clock.
  • The Sun’s rays reaching two cities are not parallel.

Safety card

Low riskLearners carry it out

Hazards

  • looking at the Sun
  • sun exposure

Controls

  • watch the shadow, never the Sun
  • hats and sunscreen

Note

No hazardous chemicals or heat sources: record the activity in the school's RiskAssess risk assessment, following the NSW Department of Education Science safety and compliance page; the Chemical Safety in Schools package is not triggered.

Curriculum references

The NSW syllabus outcomes and Australian Curriculum v9 codes this activity supports. They are references, not a verified or complete curriculum alignment.

Sources

The pages the author read to write this activity.

  1. www.nsw.gov.au/education-and-training/nesa/curriculum/science/investigating-science-stage-6-2017
  2. education.nsw.gov.au/content/dam/main-education/teaching-and-learning/curriculum/key-learning-areas/science/s-6/investigating-science/Investigating-Science-Module-5-guide.docx
  3. www.nasa.gov/wp-content/uploads/2009/07/136202main_measure.of_.the_.universe.pdf
  4. instructional-resources.physics.uiowa.edu/8a0530-eratosthenes-measurement-earths-radius
  5. geodesyapps.ga.gov.au/vincenty-inverse

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