Measuring the Earth

The Big Question: How Big is Earth?

For most of human history, no one knew how large the Earth was. Ancient peoples could see the horizon, feel the curve of a coastline, and watch ships disappear hull-first over the sea — enough to know Earth was round — but measuring it seemed impossible.

How would you measure the size of a planet you're standing on? You can't see it from space. You can't drive around it with an odometer. The answer required an extraordinary leap of imagination: using the angle of sunlight and the known distance between two cities.

Key idea: If the Sun is so far away that its rays hit Earth in parallel lines, then any difference in shadow angles between two cities must be caused by the curvature of Earth itself. Measure the angle and the distance, and you can calculate the whole circumference.

Eratosthenes — The Librarian Who Measured a Planet

Eratosthenes of Cyrene (276–194 BC) was the chief librarian of the great Library of Alexandria in Egypt — the largest collection of knowledge in the ancient world. He was a mathematician, geographer, astronomer, and poet. His contemporaries called him "Beta" (second best) because he excelled at everything but was never the single greatest in one field. However, his measurement of Earth's circumference, made around 240 BC, stands as one of the most remarkable scientific achievements in history.

The Clue in a Well

Eratosthenes read a curious report from the city of Syene (modern-day Aswan, Egypt): on the summer solstice at noon, the Sun shone straight down into a deep well — no shadow on the walls, because the Sun was directly overhead. He also noticed that on the same day, at the same time in Alexandria (about 800 km to the north), a vertical stick cast a shadow.

This bothered him. If the Earth were flat, the Sun would be directly overhead both places at noon — yet Alexandria had a shadow and Syene did not. The only explanation: the Earth is curved, and the two cities sit at different angles relative to the Sun.

The Four Steps of His Experiment

1

Identify a place where the Sun is directly overhead. Syene was known to have the Sun shine straight down a deep well on the summer solstice — meaning the Sun was at exactly 90° (directly overhead) at noon.

2

Measure the shadow angle in Alexandria on the same day. Eratosthenes placed a stick (called a gnomon) vertically in the ground in Alexandria and measured the angle of its shadow at noon on the summer solstice. He found it was 7.2° — exactly 1/50th of a full circle (360°).

3

Understand why that angle exists. Because the Sun's rays are parallel, the 7.2° difference in shadow angle equals the angle between Syene and Alexandria as seen from Earth's center. In other words, the arc between the two cities spans 1/50th of Earth's full circumference.

4

Multiply by 50. If the 800 km between the two cities equals 1/50th of the circumference, then the full circumference is 50 × 800 km = 40,000 km. The modern accepted value is 40,075 km. Eratosthenes was off by less than 2%.

Circumference = 360° ÷ Shadow Angle × Distance between cities

Eratosthenes: 360° ÷ 7.2° × 800 km = 50 × 800 = 40,000 km

What is a gnomon? A gnomon (NOH-mon) is simply a vertical stick or post used to cast a shadow and measure angles. Sundials use a gnomon. The word comes from Greek, meaning "one who knows" or "indicator." It is one of humanity's oldest scientific instruments.

Others Who Measured the Earth

Eratosthenes wasn't the last to tackle this problem. Over the next 2,000 years, scientists across the world refined the measurement using better instruments, higher mountains, and eventually satellites.

240 BC

Eratosthenes of Cyrene
~40,000 km · 0.2% off
Using shadow angles in Alexandria and Syene. He measured the arc at 7.2° and the distance at 5,000 stadia. Depending on the exact length of a stadion used, his result falls between 39,375 km and 46,250 km — with the most likely value of ~40,000 km being remarkably accurate.
100 BC

Poseidonius of Apamea
~38,600 km · 3.7% off
A Greek philosopher and polymath, Poseidonius used the star Canopus rather than the Sun. He measured the angle difference of Canopus above the horizon at Rhodes and Alexandria, then multiplied by the distance between them. His estimate ranged from 29,000 km to 38,600 km across different attempts — the latter being quite good. Ironically, his lower estimate was the one Columbus used centuries later, leading Columbus to badly underestimate how far away Asia was.
827 AD

Al-Ma'mun's Expedition (Baghdad)
~40,250 km · 0.4% off
The Abbasid Caliph Al-Ma'mun ordered two teams of surveyors to walk in opposite directions along a flat plain in Mesopotamia (modern Iraq), measuring how far they had to travel before the North Star's altitude changed by exactly one degree. The average of their results gave a degree of latitude of about 111.8 km — implying a circumference of around 40,248 km. Extraordinarily precise for the 9th century.
1030 AD

Al-Biruni of Khwarezm
~40,248 km · 0.4% off
The Islamic scholar Abu Rayhan Al-Biruni devised an entirely new method requiring only a single person and a mountain. He measured the height of a mountain, then used a special instrument to measure the dip angle of the horizon from the summit. Combining these two measurements with trigonometry, he calculated Earth's radius — and thus its circumference — to within 0.4% of the modern value. This technique required no second city and no team of surveyors.
1669

Jean Picard (France)
~40,062 km · 0.03% off
French astronomer Jean Picard was the first to use telescopic sights and triangulation to measure a meridian arc. He carefully surveyed 1.1° of latitude along the Paris meridian using a chain of triangles and measured star angles with a precision instrument. His result of 40,062 km was so accurate that Isaac Newton used it to verify his law of universal gravitation — the new precision confirmed Newton's predictions perfectly.
1799

Méchain & Delambre (France)
~40,009 km · 0.02% off
Pierre Méchain and Jean-Baptiste Delambre spent seven years (1792–1799) measuring the meridian from Dunkirk to Barcelona, covering 9.5° of latitude with unprecedented precision. Their mission was not just scientific — it was political. The French Revolutionary government wanted a universal unit of measurement based on nature. The result: the metre was defined as 1/10,000,000 of the distance from the North Pole to the equator. Every metre you measure today is a direct descendant of their survey of Earth's circumference.
Today

GPS Satellites & Laser Ranging
40,075.017 km · exact
Modern measurement uses a network of GPS satellites, ground stations, and laser ranging to track Earth's exact shape in real time. Earth is not a perfect sphere — it bulges slightly at the equator due to its rotation (equatorial circumference: 40,075 km; polar circumference: 40,008 km). The difference is 67 km — less than 0.2%. Not bad for a shadow and a stick.

How Accurate Were They?

Scientist	Year	Result (km)	Error	Accuracy
Eratosthenes	240 BC	~40,000	0.2%	99.8%
Poseidonius	100 BC	~38,600	3.7%	96.3%
Al-Ma'mun's team	827 AD	~40,250	0.4%	99.6%
Al-Biruni	1030 AD	~40,248	0.4%	99.6%
Jean Picard	1669	40,062	0.03%	99.97%
Méchain & Delambre	1799	40,009	0.02%	99.98%
GPS / Modern	Today	40,075.017	0.000%	100%

Why Measuring Earth's Circumference Matters

📏

The Metre Was Born From It The metric system — used by nearly every country on Earth — was literally defined as a fraction of Earth's circumference. One metre was set as 1/10,000,000 of the distance from equator to pole. Every ruler, every kilometre, every km/h speed limit descends from this measurement.

🧭

Navigation and Cartography Accurate maps require knowing Earth's size. Before Picard's 1669 measurement, maps had systematic errors that led ships astray. Knowing the exact circumference let navigators calculate longitude and latitude with far greater precision.

🛰️

GPS and Satellite Orbits Every GPS satellite knows Earth's exact shape and circumference. Your phone uses this to triangulate your position to within a few metres. Even the height and speed of satellite orbits are calculated from Earth's known size and gravity.

🔭

Verifying Newton's Laws When Picard measured Earth's circumference precisely in 1669, Newton used that number to check whether his theory of gravity correctly predicted the Moon's orbit. It did — confirming one of the greatest scientific theories in history.

Practice Problems

Use the formula: Circumference = 360° ÷ angle × distance. Round to the nearest whole number unless stated.

Easy1. Eratosthenes measured a shadow angle of 7.2° and a city distance of 800 km. Use his formula to calculate Earth's circumference in km.

Hint: 360 ÷ 7.2 × 800 = 50 × 800 = 40,000 km

Easy2. If you measured a shadow angle of 9° between two cities that are 1,000 km apart, what circumference would you calculate?

Hint: 360 ÷ 9 × 1,000 = 40 × 1,000 = 40,000 km

Medium3. Poseidonius measured a star angle of 7.5° between Rhodes and Alexandria (distance: 800 km). What circumference did he calculate? Round to the nearest 100 km.

Hint: 360 ÷ 7.5 × 800 = 48 × 800 = 38,400 km

Medium4. Eratosthenes' result (40,000 km) differs from the modern value (40,075 km) by how many km?

Hint: 40,075 − 40,000 = 75 km

Challenge5. One degree of latitude equals about 111 km on Earth's surface. If a student in your city measures a shadow angle of 4.3° compared to a city 478 km south where the Sun is directly overhead, how close is their calculated circumference to the real value? Calculate their result first, then find the % error to 1 decimal place. (% error = |result − 40,075| ÷ 40,075 × 100)

Hint: 360 ÷ 4.3 × 478 = 83.72 × 478 ≈ 40,018 km. Error = |40,018 − 40,075| ÷ 40,075 × 100 ≈ 0.1% — impressively close!

Challenge6. The metre was defined as 1/10,000,000 of the distance from the equator to the North Pole. Using Earth's polar circumference of 40,008 km, calculate this pole-to-equator distance in km, then check whether it equals exactly 10,000 km (one ten-millionth of that should be 1 m). Enter the distance in km.

Hint: Equator to pole = polar circumference ÷ 4 = 40,008 ÷ 4 = 10,002 km. The metre was meant to be exactly 1/10,000,000 of this (= 1.0002 m by modern standards — very close!).