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Measure the Earth with a pencil

You need sunshine, a pencil and a ruler. At your local noon, the pencil's shadow and the date show how far you are from the Equator. Two shadows, in two towns, show how big the planet is. That is how Eratosthenes measured it, more than 2,200 years ago.

București

In Bucharest on 28 September 2026 the Sun is highest at 13:06, Romanian time. In Satu Mare at 13:19; in Tulcea at 12:56. The further west, the later.

It is exactly what you do with two pencils: one angle measured, one distance taken from others.

How to measure the shadow: four steps, ten minutes

  1. A white sheet, in the sun, on something level: an outdoor table, a windowsill, the school yard. Check it with your phone's spirit level, or a glass of water: the water should stand equally high all round.
  2. The pencil, straight up. Fix it with a little modelling clay and check with a set square: one side against the pencil, the other on the paper. A pencil leaning one degree towards or away from the Sun moves the result by 0.1–0.9 degrees, depending on the season; around the equinoxes, by about half a degree, some 55 km.
  3. At the time shown above, mark the end of the shadow. The end is blurred over a few millimetres because the light comes from the whole of the Sun's disc, half a degree wide. Put the mark in the middle of the blur.
  4. Measure two lengths: the pencil, from the paper to its tip, and the shadow, from the foot of the pencil to your mark. Only the ratio between them matters, so any pencil and any ruler will do. A metre stick gives a better result.
How precisely you read the ruler

Type the length of the pencil and of its shadow.

Cloudy? The shadow can be measured on any clear day, not just today. The page works out the noon for each day.

What your shadow says: how far you are from the Equator

The Sun's rays reach the Earth practically parallel. If somewhere today the Sun stands straight overhead, a stick there casts no shadow. The further north of that place you are, the more slanting the rays and the longer the shadow.

The angle between the pencil and the rays is the same as the angle between your vertical and the vertical of the shadowless place, two lines that, on a round Earth, meet at the centre. Add it to the latitude of the shadowless place and you have your own latitude.

Your result appears here once you type the two lengths.

The angle between the pencil and the rays is the same as the angle between your vertical and the vertical of the shadowless place, two lines that, on a round Earth, meet at the centre. Add it to the latitude of the shadowless place and you have your own latitude.

Two shadows, two towns: how big is the Earth

One shadow gives you your latitude. The size of the planet needs two. If two people, one further north and one further south, measure their shadows at their own noon, the angles differ. The difference is the slice of the circle between them. If the slice is 4 degrees, the Earth is 90 times the distance between them, because 360 ÷ 4 = 90.

The angle is yours to measure. The north–south distance between your parallels comes from the map, just as Eratosthenes took his from others. Each of you measures at your own noon, so it doesn't matter if one is further east: the Sun gets highest at different times, but on the same day it stands at practically the same declination, so only latitude changes its height.

The link carries only the town, the two lengths, the day and the time. Nothing is stored anywhere; it is all in the link.

The further apart you are north to south, the more precise the result. Zimnicea and Sighetu Marmației are more than 4 degrees of latitude apart, about 475 km, nearly the full north–south span of Romania.

One shadow gives you your latitude. The size of the planet needs two. If two people, one further north and one further south, measure their shadows at their own noon, the angles differ. The difference is the slice of the circle between them. If the slice is 4 degrees, the Earth is 90 times the distance between them, because 360 ÷ 4 = 90.

In class: twenty pencils, one Earth

  1. Morning, in the classroom. Put the page on the projector and read off the time of noon for your town. Show the drawing with the parallel rays: why would sticks cast different shadows in different towns?
  2. At noon, in the yard. In pairs: a pencil or a stick, a set square, a ruler, a sheet of paper. All pairs mark the shadow at the same time and write down the two lengths.
  3. Afterwards. Enter the readings below, on the projected screen. The class median is your best estimate of your latitude. Send the class link to a school in another town: two medians give you the Earth.

In Romania this fits the grade 5 geography programme (2017): the Earth's “shape and dimensions” and “geographic coordinates”.

    There is also an international network of schools that measure on the same day, around the equinoxes: the Eratosthenes Experiment. In March 2016 more than 150 Romanian schools signed up.

    How Eratosthenes did it, according to the fullest surviving account

    Eratosthenes of Cyrene (c. 276–194 BC) ran the Library of Alexandria. The book in which he described the measurement, On the Measurement of the Earth, is lost. We know the method mainly from an astronomy handbook written by Cleomedes some centuries later; Vitruvius and others give only the result and what it was built from.

    According to Cleomedes, at Syene (today's Aswan), at noon on the summer solstice, the pointers of sundials cast no shadow: the Sun is straight overhead. At the same moment in Alexandria, the shadow in a bowl-shaped sundial covers an arc of one fiftieth of a circle, 7.2°. From Syene to Alexandria is 5,000 stades. So the Earth measures 50 × 5,000 = 250,000 stades.

    It is exactly what you do with two pencils: one angle measured, one distance taken from others.

    According to Cleomedes, at Syene (today's Aswan), at noon on the summer solstice, the pointers of sundials cast no shadow: the Sun is straight overhead. At the same moment in Alexandria, the shadow in a bowl-shaped sundial covers an arc of one fiftieth of a circle, 7.2°. From Syene to Alexandria is 5,000 stades. So the Earth measures 50 × 5,000 = 250,000 stades.

    What is not in the text, and what was approximate

    • The well. Strabo and Pliny the Elder mention a well at Syene lit to the bottom at the solstice. In the same passage Pliny puts Syene 5,000 stades from Alexandria, but neither names Eratosthenes, and Cleomedes, who describes the method, does not mention it.
    • Syene was not quite on the Tropic. At the time the Tropic of Cancer ran at about 23.74°. Aswan is at 24.09°: 0.35° further north, about 40 km.
    • Alexandria is not due north of Syene. It is 3 degrees further west. The true angle between their parallels is 7.11°, not 7.2°; the north–south distance about 790 km.
    • 250,000 or 252,000. Cleomedes gives 250,000 stades; Strabo, Pliny and Vitruvius give 252,000. Christián Carman and James Evans (2015) argue that both are Eratosthenes', as a lower and an upper bound: 250,000 if the Sun is taken as infinitely far away, 252,000 if you allow for the distance at which he put it.
    • Not the first estimate. Aristotle mentions mathematicians who gave the Earth 400,000 stades; Archimedes reports others who tried to prove about 300,000. Eratosthenes is the first whose method survives.
    • Another version. In the account of Martianus Capella, written later still, the angles are taken at Syene and at Meroë, further south, and the distance is supplied by the royal surveyors of Egypt.

    How close did he get? It depends how long a stade was

    A 157.5 m stade (Hultsch, 1882)39,690 km
    The truth, through a pole40,008 km
    A 185 m stade (Engels, 1985)46,620 km
    Eratosthenes' 252,000 stades in kilometres, under two proposed lengths of the stade

    A stade was the length of a running track, and Greek tracks were not all the same length. No surviving text says which stade Eratosthenes used, so his accuracy cannot be stated without an assumption.

    In 1882 Friedrich Hultsch proposed a stade of 157.5 m, starting from an Egyptian unit of length. With it, the 252,000 stades make 39,690 km: within 1% of the truth. In 1985 Donald Engels argued for the ordinary stade of about 185 m; that gives 46,620 km, 16.5% too much. Pliny himself converted the figure into 31.5 million Roman paces, which also makes stades of about 185 m.

    The method was sound. How accurate the result was remains a dispute among specialists. You, with a pencil and a map, can check yourselves on the spot.

    Tomorrow at noon the Sun will stand a little differently in the sky, and the shadow will have another length. Your latitude will come out the same.

    More on the site

    Sources and method

    The time of noon and the Sun's latitude (declination) come from the NOAA algorithm after Jean Meeus, Astronomical Algorithms. Against the JPL DE421 ephemeris, over 54 combinations of towns and days, the largest difference is 2 seconds in time and 0.0012° in declination. For Romanian towns, times are Romanian time; for a place outside Romania given by your phone, your phone's time.

    Your latitude = the shadow's angle + the Sun's declination at your noon. The shadow's angle is the arctangent of shadow over pencil. If you marked the shadow before or after noon, the page adds back the height the Sun has lost in the meantime; 45 minutes from noon the correction is exact to 0.001°. Atmospheric refraction, 1 to 3 minutes of arc at Romanian noon (0.05° at most), is ignored.

    The tolerance you choose applies to both lengths; the interval shown is the worst case, not counting any lean of the pencil, which adds to it. The north–south distance between two parallels is computed on the WGS84 ellipsoid; the reference circumference, 40,008 km, is the one through the poles. Coordinates are those of Romania's 3,181 administrative units (communes, towns and cities).