How To Disprove The Flat Earth


By @ScienceWasWrong

While retweeting flat earthers, as a Take That account, I was much struck with certain facts about the positions of the heavenly bodies, and in their relation to the geographic coordinates of the observer. These facts seemed to me to throw some light on the shape of the earth — that mystery of mysteries, as it has been called by one of our greatest philosophers. On my return home, it occurred to me, in 2015, that something might perhaps be made out on this question by patiently accumulating and reflecting on all sorts of facts which could possibly have any bearing on it. After five minutes work I allowed myself to speculate on the subject, and drew up some short notes; these I enlarged in late September into a sketch of the conclusions, which then seemed to me probable: from that period to the present day I have steadily pursued the same object. I hope that I may be excused for entering on these personal details, as I give them to show that I have not been hasty in coming to a decision.

eratosthenes experimentAround 240 B.C, the Greek astronomer Eratosthenes devised a way to measure the circumference of the earth. He knew that each year on the summer solstice the sun would pass directly overhead and illuminate the bottom of a well in the city of Syene, about 500 miles south of Alexandria. On that day, when the sun was at its highest in the south, he found that a vertical stick in Alexandria cast a shadow at a 7.2 degree angle. This angle corresponds to the solar zenith angle – the angle between the sun and the point in the sky directly overhead. He reasoned that the distance between the two cities must therefore constitute 7.2 degrees of a circle, which gives a circumference of about 25,000 miles.

Eratosthenes made two assumptions here: that the earth is a globe and that the sun is distant enough that its rays are effectively parallel.  Eratosthenes’ experiment alone does not prove the earth is a globe because his assumptions must be true in order for his results to be valid. Flat earthers reject these assumptions and claim that the sun is much smaller and closer to the earth so that its rays are not parallel. This can produce a 7.2 degree shadow just as easily. In order to find out which is the case, we must work backwards from these assumptions and see what each of them would entail if true. Then we can find a way to test them.


Modern science places the earth’s radius at 3,959 miles, which gives a pole to pole circumference of about 24,875 miles. Dividing by 360, we find that 1 degree of arc should be about 69.1 miles on the earth’s surface. That is why lines of latitude are about 69.1 miles apart. So on a globe for every 69.1 miles or 1 degree of latitude you move north or south of the point the sun is directly overhead, the angle of the shadow would increase by 1 degree. If Eratosthenes was correct, the angle of his shadow must have been equal to Alexandria’s latitude north of Syene, because that would be the angle that Alexandria is leaning away from Syene due to the curvature of the earth. If three other people were doing his experiment on the same day at 20 degrees, 40 degrees, and 60 degrees north or south of the Tropic of Cancer, each respectively would measure a shadow angle of 20 degrees, 40 degrees, and 60 degrees, and all would arrive at a circumference of 24,875 miles. The beauty of Eratosthenes’ experiment is that you can do it anywhere on earth, on any day of the year and arrive at the same circumference. This is where the flat earth model runs into problems.

sun angles

In the flat earth model, as a consequence of geometry, as your distance from the point the sun is directly overhead increases, so must the distance between each degree of shadow. If the sun were 3,000 miles above the flat earth, it would be 1,092 miles to the 20 degree shadow, 1,425 miles between the 20 and 40 degree shadows, and a whopping 2,680 miles between the 40 and 60 degree shadows. Someone doing Eratosthenes’ experiment 20 degrees north of the Tropic of Cancer would arrive at a circumference of 19,654 miles, someone at 40 degrees north would get 22,656 miles and someone at 60 degrees north would get 31,177 miles.

Flat earthers use the Azimuthal Equidistant Projection map, which shows all points at an undistorted distance and direction from the center. Their lines of latitude have flat earth mapthe same spacing as the globe: 69.1 miles per degree. It is for this reason that the angle of the shadow can only equal the latitude between you and the point the sun is directly overhead in one location. Everywhere south of that point, the shadow angle would be greater than the latitude and everywhere north of that point, the shadow angle would be less than the latitude. By the time the latitude catches up with the shadow angle, each degree of shadow continues to get progressively farther apart and latitude passes it. The 20 degree shadow would be 15.8 degrees north of the point the sun is directly overhead, the 40 degree shadow 36.4 degrees north, and the 60 degree shadow 75.2 degrees north. Only around 48.3 degrees north would the latitude and angle actually match up as they should everywhere on the globe and only there would someone arrive at a circumference of 24,875 miles. You can change the height of the sun all you like, the latitude and angle will only be the same in one location. Even then, it is only a coincidence due to the random height of the sun, not a direct function of latitude as it is on the globe.

So to recap, if the earth is a globe, the angle between the sun and 90 degrees overhead must be equal to the latitude between you and the point the sun is 90 degrees overhead in every location. If the earth is flat, this can only be true in one location. Testing this discrepancy between these two models has the potential to end to this debate once and for all.

You can do this test yourself. You can’t be in two places at once to take multiple measurements on the same day, but, lucky for us, the position of the sun in the sky varies by about 47 degrees every 6 months as the sun swings back and forth between the Tropic of Cancer and the Tropic of Capricorn. Since the sun is moves a few degrees north or south each day, you can stay right where you are and test a new angle every day.

  1. Go to this website, enter your location, and look at the sun transit time to find out when the sun will cross over your line of longitude, when it is 180 degrees south of you.rolling pin
  2. When this time comes, go outside, find something that casts a shadow, measure it, and carefully measure the length of the shadow. Only measure to the end of the dark part of the shadow, not the lighter penumbra around the edges. The object should be as vertical as possible and on a level surface. The taller it is, the more precise your measurement will be.
  3. Divide the length of the shadow by the height of the object and hit inverse tangent on your calculator to get the angle of the shadow. Now, the sun has an angular diameter of about 0.5 degrees. The end of the dark part of the shadow is defined by light from the top of the sun passing over the top of the object, while light from the bottom of the sun crosses over it to mark the end of the penumbra, giving it the same angular width as the sun. To get the zenith angle of the center of the sun, add 0.25 degrees to your shadow angle.penumbra
  4. Go to this website and see which latitude the sun was at zenith at the time you measured the shadow.
  5. If the sun is the same hemisphere as you and at a lower latitude, add the sun’s latitude to the angle you measured. If it is the same hemisphere and at a higher latitude, subtract the angle from the sun’s latitude. If it is in the opposite hemisphere, subtract the sun’s latitude from the angle you measured. First convert the latitudes’ arc minutes to decimals by dividing by 60. (Example: 46° 34′ = 46 + 34/60 = 46.567°)
  6. If the earth is a globe, the result, depending on the precision of your measurements, should be equal to your latitude.
  7. To find the circumference of the earth as Eratosthenes did, multiply the latitude between you and the sun by 69.1 to get the distance. Divide 360 by the angle of the shadow, and multiply that by the distance. The result should be about 24,875 miles.


  1. Find the angle of a shadow at ANY time of day when the sun is out.
  2. See where the sun was at zenith at that time.
  3. Type those coordinates into Google Earth and measure the distance from there to your location with the ruler set to degrees.
  4. Compare this to the angle you measured.

This method is especially damning because of the degree in which longitude lines diverge south of the equator on the flat earth map.

Still unconvinced by the astronomical coincidence that at a random time on a random day the sun was at the correct height and distance to be at the same angle in the sky as your random latitude north or south of it? Well it’s the moment of truth, because now the angle should get exponentially wronger with each passing day. Try it again the next day, or the next day, or the day after that. You can also do this:

  1. Multiply the angle by 69.1 to get your distance from the point where the sun was at zenith.
  2. Divide this by the tangent of the angle to get what should be the height of the sun if the earth is flat.
  3. Now see what latitude the sun will be at zenith two weeks from now. Convert it to decimals. If it’s in the opposite hemisphere, add it to your latitude, if not, subtract the lower from the higher. This is the angle of the shadow you should get if the earth is round.
  4. Multiply this angle by 69.1 to get the distance.
  5. Divide the distance by the height of the sun from step 2 and hit inverse tangent to get the angle of the shadow you would expect on a flat earth.
  6. Wait two weeks and find the angle of a shadow again as you did in the first test.

If you get the angle you found in step 3, the earth is round. If you get the angle you found in step 5, the earth is flat. Still not convinced? Try it on the winter solstice, the spring equinox, and the summer solstice. Be amazed as you get the same angle you would expect to get on a globe each time.

Is there any way for a flat earther to ad hoc their way out of this? They could say that the sun is constantly changing height to give you the correct angle. Presumably it would be doing this for your sake as it would only work for someone at your exact latitude, giving everyone else in the world wildly inaccurate angles. They could say the sun position website is wrong and the sun is moving away at increments that give you the correct angle each day, even though again, this would only be working for you. If you’re in England, the sun would have to be past the damned ice wall on the winter solstice to get an angle like 75°. They could come up with some kind of woo about the position of the sun being an illusion, but while you’re over there in David Copperfield land with your buoyancy gravity, we’ll be here in reality getting things done.

sun trigonometry


22 thoughts on “How To Disprove The Flat Earth

  1. Son of Sharecroppers October 21, 2015 / 5:43 am

    This is well done. But I think that tI have two easier responses.

    1. All of the modern flat-earthers whom I’ve encountered assume that the earth is a flat disk. They must do so because abundant evidence shows that people can travel from east to west and return to their starting point.

    If the earth is a flat disk, and if the sun and moon spin around that disk at relatively low elevations, then the angular diameter of those bodies should change as they traverse the sky. But that is not the case. Instead, both the sun and moon appear to have an angular diameter of about one-half of one degree, regardless whether they are on the horizon or directly overhead. That very simple observation (which almost anyone can make) disproves the hypothesis of a disk flat earth.

    2. Many amateur astronomers (such as I am) use telescopes that use what are called “German equatorial mounts.” A German equatorial mount consists of two elements, each of which may rotate, arranged at 90 degree angles to one another. The axis of one of the elements points directly north.

    All equatorial mounts assume that the earth is a ball. One may align the mount so that one axis points due north. With the mount correctly aligned, the telescope will then track any heavenly body from east to west without having to correct for any north-south deviation.

    It is true that the earth could stand still, with the heavens rotating around it. But it would be impossible for an equatorial mount to track without any north-south deviation if the earth were a flat disk. If the earth were a flat disk, with the heavenly bodies spinning around it, then those bodies would trace very different paths across the sky than are traced by bodies that appear to be rotating around a globular earth. And those bodies would move at different speeds, depending on their angular distance from true north. On the flat-earth theory, a celestial body near true north would appear to move more slowly than would a celestial body near the “equator.”

    Tens and even hundreds of thousands of amateur astronomers can attest to the contrary. If a telescope using an equatorial mount is set up in even an approximately correct manner, that telescope will track celestial bodies from true north (i.e., near Polaris) to as far south as the telescope may be pointed without altering the speed of tracking. This fact absolutely disproves the theory of the “flat disk earth.”

    3. All published star charts assume that the earth is a globe. If the “flat disk earth’s” were right, then the angular distance between celestial bodies would be wrong–and those apparent angular distances would change as the celestial bodies traversed the sky. But that is not the case. I can personally attest that the angular distances between celestial bodies, as identified by star charts assuming a globular earth, match observation.

    I am positive that no flat-earthers are amateur astronomers. If you mention “equatorial telescope” to any of them, they will shut up

    Liked by 2 people

    • fred ether October 4, 2016 / 5:58 pm

      dumb article…..for anyone who has individual thought. FYI


  2. Rob M (@rob_s_fl) November 10, 2015 / 6:03 am So if Okeechobee County Airport is 33.4 feet above seal level (Okeechobee, FL … Elevation: 33.4 ft.); Is that 33.4 above the Atlantic mean sea level 40 miles East or 33.4 feet above the Gulf of Mexico mean sea level 100 miles West or 33.4 feet above sea level at the southernmost point in Key West 190 miles Southwest?

    Liked by 1 person

    • planateearth December 29, 2015 / 7:22 pm

      Rob M I suppose no one could wrap their mind around your obviously intelligent question!

      Liked by 1 person

    • John April 21, 2016 / 3:54 am

      What is your point? On a round earth or other similar styled object, how far should you be able to see? I can see the beach of an island 10 miles away on whatever we are on. Please explain that on your map.


      • Chris February 20, 2017 / 7:17 am

        Anyone in new Zealand can see around 100 miles from beach to beach and there’s no curve on the flat ocean sea level


    • John April 21, 2016 / 3:56 am

      Most of Florida is below sea level.


  3. sindre February 27, 2016 / 4:35 pm

    you are wrong, earth is flat
    Eratosthenes’ stick experiment can not only tell us about the size of the earth, but can also be used to compute the distance to the sun as well. If the earth is round, the celestial bodies are computed to be millions of miles distant. If the earth is flat, the celestial bodies are triangulated to be relatively close to the earth’s surface.

    In his experiment Eratosthenes assumes that the earth is a globe and that the sun is very far away in his computations for the size of the earth and the distance to the sun. However, if we use his data with the assumption that the earth is flat we can come up with a wildly different calculation for the distance of the sun, showing it to be close to the earth. The sun changes its distance depending on the model of the earth we assume for the experiment.

    Millersville University goes over the two ways of interpreting Eratosthenes’ data. The first part of the article goes over the interpretation of his data under a Round Earth model, and the bottom part of the article goes over an interpretation of the data under a Flat Earth model.

    Eratosthenes’ model depends on the assumption that the sun is far away and therefore produces parallel rays of light all over the earth. If the sun is nearby, then shadows will change length even for a flat earth. A flat earth model is sketched at the right. The vertical stick casts shadows that grow longer as the stick moves to the left, away from the closest point to the sun. (The sun is at height h above the earth.)

    A little trigonometry shows that


    Using the values 50 degrees and 60 degrees as measured on the trip, with b=1000 miles, we find that h is approximately 2000 miles. This relatively close sun would have been quite plausible to the ancients.

    Continuing the calculation, we find that a is approximately 2400 miles and the two distances R1 and R2 are approximately 3000 and 3900 miles, respectively.”


    • TakeThatAccounts February 29, 2016 / 5:07 am

      Did you even read the entire post, or did you just get to the word Eratosthenes before you started copying and pasting Flat Earth Wiki? The two models make widely different predictions about the specific angle of the shadow you would expect. If the earth is a globe, the angle of the shadow should be equal to the degrees of arc between you and the point the sun is 90 degrees overhead, any time of day, any day of the year. On a flat earth this is simply impossible. I spent paragraph after paragraph explaining this. Go test it for yourself.

      Liked by 1 person

      • tony March 27, 2016 / 5:10 pm

        I’d point out that the very last line of the reference cited ( says, “We conclude that the flat earth/near sun model does not work.”

        Liked by 3 people

    • Saul Trane April 2, 2016 / 7:19 pm

      You obviously don’t understand the above article. It’s clearly demonstrating to you that the sun’s angles which we observe on a spherical earth cannot be produced on a flat earth. It’s geometrically impossible.

      Liked by 2 people

  4. Chris Miller May 24, 2016 / 12:03 pm

    Some of the comments in this Blog explain perfectly why there are people that believe that Flat Earth is possible given the vast amount of evidence. You can show them, explain to them, give them experiments to try and they just do not have the mental capability to process the information accurately. They fall back to the easiest explanation because of the own mental laziness. It is Sad to see.

    Liked by 1 person

  5. BillyJoe August 15, 2016 / 7:33 am

    The earth can not be round because Jesus.
    Nowhere in the scripture is a ball shaped earth mentioned. But indeed, the Lord stopped the sun for a day to give Izrael more light to fight a battle.
    If the earth was round, all planes would have to constantly drop by eight miles per inch squared travelled.
    And all that inverted tangentials you are talking about is just bullshit that the government put into the calculaters to confuse us. I have borrowed cousin Billybob’s calculater and tried this inverted tangen and there was not a single number I would understand coming out of that thing. Not even that stupid 69.1 you are talking about.

    This whole globe thing is just a theroy, just like evolution. It’s a theory, not a fact.


    • Chris C September 28, 2016 / 8:28 am

      Billyjoe, it’s obvious that you’re not too bright. I’m sure that even basic physics/geometry is beyond your realm of understanding.


      • David January 17, 2017 / 4:48 am

        The same could be told about you, Chris C. Surely the definition of the word “irony” is outside of your reach.


  6. victoria November 13, 2016 / 12:52 am

    I am here to say not all flat earth people believe the flat disk model with the sun and moon circling over. This model just doesn’t work with observable facts and the existence of equatorial sun dials proves the sun follows a straight path. But the absence of an observable arc is compelling. You won’t find a curve anywhere on earth (this is the real science). Also, the curve formulas don’t seem to work either. Were these just planted in science books as busy work for students? I believe the earth is flat, but probably not a flat disk. That’s been a bit of disinformation.


    • TakeThatAccounts November 14, 2016 / 5:21 am

      The curve formula “8 inches per mile squared” was first conceived of by a flat earther in 1865, and despite the fact that it describes a parabola rather than a circle, it is still used today exclusively by flat earthers who can’t do trigonometry.


      • daznez January 19, 2017 / 12:13 pm

        nope, even using this calculator, the surveyor’s favourite – – there is still no observable curvature anywhere – or lighthouses wouldn’t work.

        islands seen from hundreds of miles away disproves the oft-tepeated baller’s argument ‘ships disappear hull first’ becuse of the curve of the ocean,at only 3-5 miles away.

        i repeat what victoria said, not all flat earthers believe in the ae circle/ gleason’s map, but we still know the eath is not moving, and it’s not just that ‘you can’t feel it because the atmosphere rotates along with it.’ what, all 6200 miles of it, all perfectly in sync? as if it’s a solid body and not made up of free-flowing gases? how DO airplanes and birds ever fly against this 1038 mph onrush of headwind if they want to travel west?

        i’ll wait.


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