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

A flat earth


How can we measure the stars if we can’t reach out and touch them?
The Earth and Moon. Image Credit: NASA/ESA
In the formative years of astronomy there was a burning desire to measure the size and distance to objects in our solar system. This tutorial will explore the following 5 questions using nothing but naked eye astronomy, geometry and trigonometry:
  • How big is the earth?
  • How big is the moon?
  • How far away is the moon?
  • How big is the sun?
  • How far away is the sun?
​Let's warm up with a simple yet powerful question: how can you prove the earth is not flat using shadows?

Is the earth flat?

A flat earth was once a commonly held belief, it persisted in some areas until the middle ages. Definitive proof of a spherical earth did not come until  Ferdinand Magellan successfully traveled around the earth (1519-1521). However this belief was challenged much earlier thanks to major advancements in ancient Greek astronomy.
The image depicts a man crawling under the edge of the sky, as if it were a solid hemisphere. This image first appeared in Camille Flammarion's L'atmosphère: météorologie populaire (1888) Image Credit: Heikenwaelder Hug
Plato (427–347 BC) was convinced that the earth was spherical: "My conviction is that the earth is a round body in the centre of the heavens...Also I believe the Earth is very vast". Significantly, he could not prove this was the case. It was his prized student Aristotle who began offering evidence for a spherical earth. One of his more convincing pieces of evidence was the shape of the earth’s shadow visible during a Lunar Eclipse. Notice the subtle curvature of the shadow cast on the moon:
Another way to prove that the earth is spherical came from Eratosthenes (276–194 BC). His method involves looking at the shadows cast by the sun on the surface of the earth. This is the basis for a sundial:
A sundial. Image Credit: John Carmichael
But what can shadows tell us about the earth? Eratosthenes did an interesting experiment using two vertical poles located in two distant cities in Egypt. You could do this using telephone poles separated by several hundred kilometers or more. (phone a friend!)
Image Credit: Brit Cruise
First wait until the sun is directly overhead point A, which is the moment when the pole casts no shadow. At the same time have a friend observe an identical pole at point B. You will find out that the pole at point B will cast a shadow. Since we can assume the sun's rays reach the earth in parallel lines this is visual proof that the surface must be curved!
Try out the interactive illustration below. You can click and drag the pole to see how it affects the shadow it casts.
Pause now and think about how these shadows could allow you to determine the radius of the earth. We have a simulation next which will help. You will need to understand how to calculate arc length so let's review that too. Have fun!

Want to join the conversation?

  • hopper cool style avatar for user ☣Ƹ̵̡Ӝ̵̨̄Ʒ☢ Ŧeaçheя  Simρsoɳ ☢Ƹ̵̡Ӝ̵̨̄Ʒ☣
    Can we calculate arc length the same way we would circumference, and just multiply by the fraction of the circle represented by the arc, or angle? T.S.
    (23 votes)
    Default Khan Academy avatar avatar for user
  • male robot hal style avatar for user chinmaybho4151
    Why do people still believe that the Earth is flat?
    (7 votes)
    Default Khan Academy avatar avatar for user
    • duskpin seedling style avatar for user EC S
      As this (https://phys.org/news/2019-01-people-earth-flat.html) site says, these people are using "Cartesian doubt." According to this philosophy, "the world outside itself is subject to uncertainty." It's mainly just a simple yet severe case of confirmation bias, exacerbated by the internet. The people who are convinced that the earth is flat search online for the five percent of evidence that supports their beliefs—and ignore the 95 percent that disproves their beliefs. Once they find this bit of information, they cling to it. As they continue searching for this information, search recommendations pop up in their feed.

      They click on these suggestions and dig deeper and deeper, further fueling their beliefs. By doing this, they discover a hub of other people who share their beliefs, and it becomes a snowball. Humans are known to gravitate towards people who share their beliefs, because it makes them feel like their beliefs are validated. These people are now so surrounded by the flat earth theory, that it becomes reality for them. Eventually, you end up with a community called, "flat-earthers.”
      (5 votes)
  • piceratops ultimate style avatar for user Joaquin Fernandez
    The flat earth myth was popularized by Washington Irving in his books. Christopher columbus did not think the earth was flat. He was catholic and in the bible there is a verse saying that the sun moves in a circuit. Any one with any sense of logic notices that ships seem to sink into the sea. that's because the earth is a sphere. What do you think?
    (3 votes)
    Default Khan Academy avatar avatar for user
  • male robot donald style avatar for user Venkat A
    OK. You all know that shape of the earth is geoid as u may have read it in books.
    Can anyone prove me that the earth is a geoid?
    (2 votes)
    Default Khan Academy avatar avatar for user
    • male robot hal style avatar for user Charles Breiling
      There are a great many YouTube videos being produced which purport to "prove" the Earth is flat, or at least that it's not a globe. But you can prove it yourself, using only careful observations of the sun moon and stars. Here's an easy one: The angle of elevation of Polaris (North star) will perfectly match your North Latitude. Only possible on a Globe. The angle of elevation of the sun at solar noon on the Equinox will perfectly match (90 minus your latitude). Only possible on a globe.

      You can even go further and discover the 23.4 degree tilt of the Earth: just measure the angle of elevation of the sun on the December Solstice and the June Solstice, and you'll find it to be 23.4 degrees southward and 23.4 degrees northward, respectively, when compared to the equinox.

      There are a ton more observations you can do. Check out the YouTube channel "Flat Earth Math" for experiments/observations you can do yourself.
      (4 votes)
  • leaf green style avatar for user atma sharan
    want to see an animation of the plane of rotation of planets around the sun. Are all in the same plane? Are their speeds comparable? Do they all go round in the same direction?
    (2 votes)
    Default Khan Academy avatar avatar for user
    • blobby purple style avatar for user Cosmos
      I found an animated view of the Solar System on Wiki. You can view the four inner planets here, but note that only Earth, Mars, and the Sun are labeled. Mercury is the sphere orbiting closest to the Sun, and Venus is the 2nd planet (between Mercury and Earth). http://upload.wikimedia.org/wikipedia/commons/0/0d/Ecliptic_plane_3d_view.gif

      Please note that I couldn't find an animation of the Outer Planets (Sorry) :/
      Answering your first question, the planets are not in the same place. Their orbits vary greatly in both distance and in time. Here's a handy article that tells the time, distance, etc. of each planet's orbit around the sun. http://www.qrg.northwestern.edu/projects/vss/docs/space-environment/3-orbital-lengths-distances.html
      (That article should also cover your question about the speed and comparability of their orbits).

      As for your final question, yes, all major planets orbit the sun in the same direction. If you were to look 'down' on the solar system so that Earth's north pole faced you, the planets would appear to orbit the sun in a counter-clockwise direction.
      Hope this helps!
      (3 votes)
  • blobby green style avatar for user Bob Cloonan
    So then why is the horizon at eye level, 180 deg all around us, no matter to our altitude? Where is the curvature??
    (2 votes)
    Default Khan Academy avatar avatar for user
  • blobby green style avatar for user Supriya Suryawanshi
    ummmm kk idk about trigonotry
    (2 votes)
    Default Khan Academy avatar avatar for user
  • starky seedling style avatar for user Teig
    How would you prove that something is 4D or 5D even?
    (2 votes)
    Default Khan Academy avatar avatar for user
  • marcimus red style avatar for user xozara
    WOW! so it said that Aristotle was the one that proved the earth to be round/spherical. however, wasn't he the one who also said that the sun orbited around the earth, which is not true? SO basically e was right about the roundness of the earth but not about the sun orbiting the earth.......:)?
    (3 votes)
    Default Khan Academy avatar avatar for user
    • purple pi purple style avatar for user Johnny Holzer
      He was one of many who believed in the geocentric model. Observations at the time left room for 'creative' orbits. You can see things on, and from, Earth that indicate it is round, but at the time, the orbits were mainly speculative. One of the first videos in this topic shows how the orbits whould have been shaped, and even lets you choose which planet to have the rest of the system orbit.
      (0 votes)
  • aqualine ultimate style avatar for user Jules Struble
    how would earth's oblate spheroid shape mess with these calculations? If the formulas assume a perfect circle, does that mean the math is off a little bit?
    (2 votes)
    Default Khan Academy avatar avatar for user