If you're seeing this message, it means we're having trouble loading external resources on our website.

If you're behind a web filter, please make sure that the domains *.kastatic.org and *.kasandbox.org are unblocked.

Main content

Eratosthenes of Cyrene

Measuring the Circumference of the Earth

More than 2,000 years ago Eratosthenes compared the position of the Sun’s rays in two locations to calculate the spherical size of the Earth with reasonable accuracy.
A Geographer and a Poet
An undated illustration of scholars at the Library of Alexandria © Bettmann/CORBIS
Eratosthenes was born in the Greek colony Cyrene, now the city of Shahhat, Libya. As a young man, he traveled to Athens to pursue his studies. He returned to Cyrene and made such a name for himself in scholarly endeavors that the Greek ruler of Egypt brought him to Alexandria to tutor his son. When the chief librarian of the famous Library of Alexandria died in 236 BCE, Eratosthenes was appointed to the prominent position around the age of 40.
A man of many talents, Eratosthenes was a librarian, geographer, mathematician, astronomer, historian, and poet. His friends at the library nicknamed him Pentathlos, or athlete who competes in five different events. The name seemed to fit a scholar who excelled in many fields of study. Most of Eratosthenes’s writings have been lost, but other scholars reported his work and findings — which were extensive.

Studying the earth

Eratosthenes may have been the first to use the word geography. He invented a system of longitude and latitude and made a map of the known world. He also designed a system for finding prime numbers — whole numbers that can only be divided by themselves or by the number 1. This method, still in use today, is called the “Sieve of Eratosthenes.”
Eratosthenes was also the first to calculate the tilt of the Earth’s axis, which he figured with remarkable accuracy; the finding was reported by Ptolemy (85-165 CE). Eratosthenes also calculated the distance from the Earth to the Moon and to the Sun, but with less accuracy. He made a catalog of 675 stars. He made a calendar with leap years and laid the foundation of chronology in the Western world by organizing the dates of literary and political events from the siege of Troy (about 1194–1184 BCE) to his own time.
Yet his most lasting achievement was his remarkably accurate calculation of the Earth’s circumference (the distance around a circle or sphere). He computed this by using simple geometry and trigonometry and by recognizing Earth as a sphere in space. Most Greek scholars by the time of Aristotle (384–322 BCE) agreed that Earth was a sphere, but none knew how big it was.
How did Greek scholars know the Earth was a sphere? They observed that ships disappeared over the horizon while their masts were still visible. They saw the curved shadow of the Earth on the Moon during lunar eclipses. And they noticed the changing positions of the stars in the sky.
A reconstruction of Eratosthenes’s c. 194 BCE map of the world, from E.H. Bunbury’s 1883 A History of Ancient Geography among the Greeks and Romans from the Earliest Ages till the Fall of the Roman Empire, public domain

Measuring the earth

Eratosthenes heard about a famous well in the Egyptian city of Swenet (Syene in Greek, and now known as Aswan), on the Nile River. At noon one day each year — the summer solstice (between June 20 and June 22) — the Sun’s rays shone straight down into the deep pit. They illuminated only the water at the bottom, not the sides of the well as on other days, proving that the Sun was directly overhead. (Syene was located very close to what we call the Tropic of Cancer, 23.5 degrees north, the northernmost latitude at which the Sun is ever directly overhead at noon.)
Eratosthenes erected a pole in Alexandria, and on the summer solstice he observed that it cast a shadow, proving that the Sun was not directly overhead but slightly south. Recognizing the curvature of the Earth and knowing the distance between the two cities enabled Eratosthenes to calculate the planet’s circumference.
Eratosthenes could measure the angle of the Sun’s rays off the vertical by dividing the length of the leg opposite the angle (the length of the shadow) by the leg adjacent to the angle (the height of the pole). This gave him an angle of 7.12 degrees. He knew that the circumference of Earth constituted a circle of 360 degrees, so 7.12 (or 7.2, to divide 360 evenly by 50) degrees would be about one-fiftieth of the circumference. He also knew the approximate distance between Alexandria and Syene, so he could set up this equation:
Eratosthenes estimated the distance from Alexandria to Syene as 5,000 stadia, or about 500 miles (800 kilometers). He made this estimation from the time it took walkers, who were trained to measure distances by taking regular strides, to trek between the cities. By solving the equation, he calculated a circumference of 250,000 stadia, or 25,000 miles (40,000 kilometers).
A diagram showing how Eratosthenes measured the Earth, accessed from Simon Fraser University Online
Several sources of error crept into Eratosthenes’s calculations and our interpretation of them. For one thing, he was using as his unit of measure the Greek unit “stadion,” or the length of an athletic stadium. But not all stadiums were built the same length. In Greece a stadion equaled roughly 185 meters (607 feet), while in Egypt the stadion was about 157.5 meters (517 feet). We don’t know which unit Eratosthenes used. If he used the Greek measure, his calculation would have been off by about 16 percent. If he used the Egyptian one, his error would have been less than 2 percent off the actual Earth’s circumference of 24,860 miles (40,008 kilometers).
A century after Eratosthenes, the Greek astronomer Posidonius of Rhodes (c. 135–51 BCE) calculated the Earth’s circumference. Posidonius used the star Canopus as frame of reference: when the star is visible at the horizon in Rhodes, it is 7.5 degrees above the horizon in Alexandria. His first calculations came out almost exactly correct, but he revised the distance between Rhodes and Alexandria, which resulted in a number comparable to about 18,000 miles (about 29,000 kilometers), some 28 percent smaller than the actual circumference. Ptolemy reported the calculations of Posidonius instead of those of Eratosthenes, and it was Ptolemy’s writings that found their way to Christopher Columbus. If Ptolemy had used Eratosthenes’s larger, more accurate figure for Earth’s circumference, Columbus might never have sailed west.
Eratosthenes lived to be about 82 years old, when he starved himself to death because he feared the onset of blindness.
By Cynthia Stokes Brown

For Further Discussion

Think about the following and share your ideas in the Questions Area below. If you were living in Greece at the time of Eratosthenes, how do you think you would have reacted to his proof? If you had believed that the Earth was flat, do you think you would have been convinced by what he was able to show?

Sources

Eratosthenes Project. Accessed June 13, 2011. http://eaae-astronomy.org/eratosthenes/.
Lasky, Kathryn. The Librarian Who Measured the Earth. New York and Boston: Little, Brown, 1994.
Nicastro, Nicholas. Circumference: Eratosthenes and the Ancient Quest to Measure the Globe. New York: St. Martin’s, 2008.
Teacher’s Guide: The Eratosthenes Project. Accessed June 13, 2011. http://www.physics2005.org/projects/eratosthenes/TeachersGuide.pdf.

Want to join the conversation?

  • leaf orange style avatar for user Jon Donym
    How come that the greeks already figured out that the earth was a sphere, while during the reneissance most people still believed the earth was a disk ?
    Did the knowledge get lost with the fall of the greek empire ?
    (2 votes)
    Default Khan Academy avatar avatar for user
    • piceratops ultimate style avatar for user trek
      The concept of the spherical earth was gradually accepted during the Middle Ages (before the Renaissance) - at least among the more educated members of the population. In fact, Saint Bede was born in the seventh century and his treatise On the Reckoning of Time makes claims about seasonal changes in daylight resting clearly upon the premise of a spherical planet.

      I think sometimes it is hard to imagine a world in which most of the population was illiterate but for most of the history of mankind, it was a reality: only in recent centuries was more than a small minority of the population able to read. Since the written word is the way we transmit accumulated knowledge across generations, it took time for knowledge to "trickle down" to the largest segments of the population.
      (14 votes)
  • mr pink red style avatar for user WallAvi
    So....does that mean that.....if the well had not been located exactly where it was that he would not have been able to calculate the circumference?

    Was this the first time the equation had been used to find the circumference/ distance -- or were the mathematicians of his day measuring circumference of things for other things such as architecture etc. And one day old Erat would have thought.....hmmmm......I wonder if I could figure out the circumference of the earth? --or was this a question being actively being worked on? Was he scoffed at for his discovery or was it generally accepted and then.....what was it actually good for in his day other than just knowing? Did it actively help cartographers, travelers or the shipping trade or did it make any sort of impact on the knowledge during and around the time that it was discovered or who would have cared enough to do something with this knowledge?

    Does anyone know how old was the well and who built it, how deep it was, how long it took to dig and whether it was intentionally placed to line up so perfectly or was it an ingredient of a Mimi goldilocks moment?
    (5 votes)
    Default Khan Academy avatar avatar for user
    • aqualine ultimate style avatar for user Adam Warren
      While the well being at a spot where the shadow was negligible was convenient, he could have used two sticks a known north/south distance apart (east/west wouldn't have been useful) and used the change in angle in place of the 7.12 degrees mentioned in the article to find the circumference. The rest you'd probably have to consult a historian sorry.
      (2 votes)
  • duskpin tree style avatar for user TicciToby #PamalaWeek
    Did he have a wife or kids?
    (3 votes)
    Default Khan Academy avatar avatar for user
  • blobby green style avatar for user atubeo123
    why did Ptolemy, not consider Eratoathenes measurement but rather Poisdonus' measurement.
    (1 vote)
    Default Khan Academy avatar avatar for user
  • blobby green style avatar for user Emily Stork
    According to space.com,(which is the source Google uses) the earth's circumference is 24,901 miles. According to National Geographic it is 24,902 miles and according to your article it was significantly less than both of those estimates. Is your information updated?
    (1 vote)
    Default Khan Academy avatar avatar for user
  • blobby green style avatar for user danielesanchez1972
    Eratosthenes

    The circumference of the Earth was already calculated in ancient Egypt and it turned out to be rather accurate considering the conditions of the time.
    http://www.mozaweb.com/Extra-Videos-Eratosthenes-209501
    (1 vote)
    Default Khan Academy avatar avatar for user
  • blobby green style avatar for user adam huerena
    how do you make a catalog for starts and how would u know if that is the same star as before?
    (1 vote)
    Default Khan Academy avatar avatar for user
  • leaf green style avatar for user James Hicks
    His figures would have convinced me that the earth was round mainly due to the fact that the curvature of the earth could be proven at the time, so why would others still believe anything but proven facts?
    (1 vote)
    Default Khan Academy avatar avatar for user
  • leaf yellow style avatar for user Lünee Vrümee #TYllwLf
    I probably would just believe that the earth's surface is just uneven, and the earth is actually flat, and also that the pole was not oriented correctly.
    (1 vote)
    Default Khan Academy avatar avatar for user
  • blobby green style avatar for user Hunter_of_Hearts
    how did they exchange the information about stick Shadow at the same time ?
    i mean they did not have a phone to tell that "hey, now there is no shadow on my obelisk"
    and the other guy is like "no waaay, we still got shadow coming from our obelisk ... how is this possible!"
    (1 vote)
    Default Khan Academy avatar avatar for user