In around 240 BC, Eratosthenes calculated the circumference of the Earth. The diagram above (from NOAA) shows how he did it. This Christmas, people in the Southern Hemisphere can repeat his work!

Eratosthenes knew that, at the summer solstice, the sun would be directly overhead at Syene (on the Tropic of Cancer) and would shine vertically down a well there. He also knew the distance to Syene.

On 21 December, the sun will be directly overhead on the Tropic of Capricorn at local noon. This table show the time of local noon on 21 December 2017, and the distance to the Tropic of Capricorn, for some Southern Hemisphere cities:

City |
Local Noon |
Distance to Tropic (km) |

Adelaide |
13:14 |
1270 |

Auckland |
13:19 |
1490 |

Brisbane |
11:46 |
450 |

Buenos Aires |
12:52 |
1240 |

Darwin |
12:45 |
1220 |

Hobart |
13:09 |
2160 |

Johannesburg |
12:06 |
310 |

Melbourne |
13:18 |
1590 |

Perth |
12:15 |
940 |

Santiago |
13:41 |
1110 |

Sydney |
12:53 |
1160 |

At exactly local noon, Eratosthenes measured the length (*s*) of the shadow of a tall column in his home town of Alexandria. He knew the height (*h*) of the column. He could then calculate the angle between the column and the sun’s rays using (in modern terms) the formula θ = arctan(*s* / *h*).

You can repeat Eratosthenes’ calculation by measuring the length of the shadow of a vertical stick (or anything else you know the height of), and using the **arctan** button on a calculator. Alternatively, the table below show the angles for various shadow lengths of a 1-metre stick. You could also attach a protractor to the top of the stick, run a thread from the to of the stick to the end of the shadow, and measure the angle directly.

The angle (θ) between the stick and the sun’s rays will also be the angle at the centre of the Earth (see the diagram at top). You can then calculate the circumference of the Earth using the distance to the Tropic of Capricorn and the fact that a full circle is 360° (the circumference of the Earth will be *d* × 360 / θ, where *d* is the distance to the Tropic of Capricorn).

Height (*h*) |
Shadow (*s*) |
Angle (θ) |

1 |
0.02 |
1° |

1 |
0.03 |
2° |

1 |
0.05 |
3° |

1 |
0.07 |
4° |

1 |
0.09 |
5° |

1 |
0.11 |
6° |

1 |
0.12 |
7° |

1 |
0.14 |
8° |

1 |
0.16 |
9° |

1 |
0.18 |
10° |

1 |
0.19 |
11° |

1 |
0.21 |
12° |

1 |
0.23 |
13° |

1 |
0.25 |
14° |

1 |
0.27 |
15° |

1 |
0.29 |
16° |

1 |
0.31 |
17° |

1 |
0.32 |
18° |

1 |
0.34 |
19° |

1 |
0.36 |
20° |

1 |
0.38 |
21° |

1 |
0.4 |
22° |

1 |
0.42 |
23° |

1 |
0.45 |
24° |

1 |
0.47 |
25° |

1 |
0.49 |
26° |

1 |
0.51 |
27° |

1 |
0.53 |
28° |

1 |
0.55 |
29° |

1 |
0.58 |
30° |

1 |
0.6 |
31° |

1 |
0.62 |
32° |

1 |
0.65 |
33° |

1 |
0.67 |
34° |

1 |
0.7 |
35° |

1 |
0.73 |
36° |

1 |
0.75 |
37° |

1 |
0.78 |
38° |

1 |
0.81 |
39° |

1 |
0.84 |
40° |

1 |
0.87 |
41° |

1 |
0.9 |
42° |

1 |
0.93 |
43° |

1 |
0.97 |
44° |

1 |
1 |
45° |

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