How Eratosthenes Measured Earth With a Stick: The 7.2° Shadow That Sized a Planet

The observation: a Sun that stood straight overhead

Eratosthenes, the chief librarian of the great Library of Alexandria, heard a remarkable report about a town far to the south called Syene (modern Aswan, on the Nile near the Tropic of Cancer). On one day of the year — the summer solstice, around June 21 — the noon Sun shone straight down a deep well there, lighting the water at the bottom, and a vertical stick cast no shadow at all. The Sun stood exactly at the zenith.

This only happens where the Sun can climb directly overhead, which on Earth occurs between the Tropic of Cancer (23.4° N) and the Tropic of Capricorn (23.4° S). Syene sits almost precisely on the Tropic of Cancer, so at the solstice the Sun reaches its northernmost point and passes through the zenith for observers there. That coincidence is the entire foundation of the experiment — Syene provided a natural "zero-degree" reference against which any shadow elsewhere could be compared.

Eratosthenes' insight was to ask a simple question: on that same day and hour, does a stick in Alexandria cast a shadow? It does. And the length of that shadow, he realized, encodes the curvature of the Earth itself.

The mechanism: why a shadow reveals curvature

The trick rests on one assumption Eratosthenes made explicitly: the Sun is so far away that its rays arrive at Earth effectively parallel. This is true — the Sun is about 150 million km away, so its rays diverge by an utterly negligible amount over the ~800 km between the two cities.

If the Earth were flat, parallel sunlight would cast identical shadows everywhere at the same instant, and every vertical stick would agree. But the Earth is curved. A stick planted at Alexandria points "up" in a slightly different direction than a stick at Syene, because local vertical follows the planet's surface. The two sticks are not parallel — they splay apart by exactly the angle subtended at Earth's center between the two cities.

Here is the elegant part. A gnomon (a vertical rod) in Alexandria at solstice noon cast a shadow whose angle from vertical Eratosthenes measured as 7.2°. By a theorem of geometry — alternate angles formed when parallel lines (the sunbeams) cross a transversal — that 7.2° shadow angle is equal to the central angle between Syene and Alexandria as seen from Earth's core. So the arc of ground separating the cities corresponds to a 7.2° slice of the whole globe.

• Syene: Sun at zenith, shadow angle = 0°.
• Alexandria: shadow angle = 7.2° from vertical.
• Difference: 7.2°, equal to the central angle between the two cities.

The arithmetic: turning 7.2° into a planet

The rest is a proportion any student can follow. A full circle is 360°. The angle Eratosthenes found was 7.2°, and:

• 360° ÷ 7.2° = 50

So the distance from Syene to Alexandria is exactly 1/50 of Earth's full circumference. Multiply that distance by 50 and you have measured the planet.

Eratosthenes took the north–south distance between the cities as roughly 5,000 stadia. (In the more refined version handed down to us he used 5,040 stadia, which conveniently gives a round total of 252,000 stadia — a number divisible by 60, yielding a tidy 700 stadia per degree.) The calculation is therefore:

• 5,040 stadia × 50 = 252,000 stadia

How he knew the distance is itself worth noting. Egypt's flat, road-linked Nile valley was routinely paced by bematists — professional surveyors trained to walk with a metronomic, equal-length stride specifically to measure distances. The regular caravan and administrative traffic between Alexandria and Syene meant the figure was reasonably well established. This is the unglamorous half of the achievement: a brilliant angle is useless without a trustworthy baseline.

How accurate was it? The stadion problem

To check Eratosthenes against a modern globe we need to know how long a stadion was — and here honesty is required, because we don't know for certain. There was no single standardized stadion in the ancient Mediterranean; estimates for the unit he used cluster between about 155 and 160 meters.

Take the commonly cited "Egyptian" stadion of 157.5 m:

• 252,000 stadia × 157.5 m = 39,690 km

Compare that to the true meridional (polar) circumference of 40,008 km and the error is under 1% — astonishing. Using the fuller plausible range of 155–160 m per stadion, the result lands somewhere between about 39,060 km and 40,320 km, an error of roughly −2.4% to +0.8%. Either way, a person with a stick got within a couple percent of a value we now nail with satellites.

It is worth being candid that this accuracy involves some luck. Eratosthenes' inputs contained real errors that partly canceled: Alexandria and Syene are not exactly on the same meridian (Syene is about 3° of longitude to the east), Syene is not precisely on the Tropic of Cancer, and the paced distance was approximate. The method is sound and the reasoning flawless; the near-perfect number owes a little to fortunate cancellation of opposing mistakes.

Common misconceptions to retire

Myth 1: Eratosthenes proved the Earth is round. He did not — educated Greeks had accepted a spherical Earth for centuries. Aristotle (~350 BC) already argued for it from the round shadow Earth casts on the Moon during a lunar eclipse and from stars shifting as you travel north or south. Eratosthenes assumed the sphere and measured how big it is. His contribution is quantitative geodesy, not the shape itself.

Myth 2: Columbus's crew feared sailing off a flat Earth. Nobody serious thought the Earth was flat in 1492. The real dispute was over size: Columbus used a badly undersized circumference (partly derived from Ptolemy and Posidonius, who had revised Eratosthenes downward) to argue Asia was reachably close. Had he trusted Eratosthenes' larger, more correct globe, he would have known an unknown continent — or an impossibly long ocean — lay in the way.

Myth 3: It required advanced instruments. The core tools were a vertical stick (gnomon) and knowledge of a well. Eratosthenes may have used a scaphe — a hemispherical sundial bowl with a central pointer — to read the shadow angle cleanly. The genius is in the geometry, not the hardware. Anyone with two locations at different latitudes, a clear solstice sky, and a protractor can repeat it today; classrooms worldwide do exactly this every June.

Why it still matters: the first measured planet

Eratosthenes (c. 276–194 BC) was a polymath — the same man built the sieve of Eratosthenes for finding prime numbers, sketched a system of latitude and longitude, and estimated the tilt of Earth's axis. Nicknamed Beta by rivals for allegedly being second-best at everything, he was in truth first at connecting them. His measurement is celebrated as the first known scientific calculation of the size of the Earth.

What makes it a landmark is not the answer but the method: a testable, quantitative model of nature built from a minimal observation and geometry, with every assumption stated. It is the template modern science still follows — and it scales. The exact same reasoning, refined, underlies:

• Al-Ma'mun's expedition (c. 827 AD), whose astronomers re-measured a degree of latitude on the Sinjar plain of northern Mesopotamia.
• The 18th-century French Academy expeditions to Lapland and Peru, which by measuring meridian arcs at different latitudes proved Earth bulges at the equator — later confirmed as 40,075 km around the equator versus 40,008 km through the poles (the polar, or meridional, circumference).
• Modern satellite geodesy, from GPS to the GRACE gravity-mapping mission, which now measures Earth's shape to the centimeter.

Every one of those is a descendant of a man watching a shadow at noon. Two and a half millennia later, the pale-blue-dot photograph from Voyager 1 in 1990 showed the whole planet as a single point of light — but it was Eratosthenes who first put a number to that dot.

Eratosthenes' figure versus the modern measured Earth

Quantity	Eratosthenes (~240 BC)	Modern value
Circumference	252,000 stadia ≈ 39,700 km	40,008 km (meridional/polar)
Equatorial circumference	not distinguished	40,075 km
Method	Solstice shadow + measured distance	Satellite geodesy (GRACE, GPS)
Angle used	7.2° = 1/50 of a circle	Confirmed by direct survey
Fractional error	≈ 2% (depends on stadion length)	reference (0%)