A direct test of the flat earth model: flight times
More data supporting a spherical earth
Published: 6 December 2016 (GMT+10)

After receiving such a surprising number of negative comments on our flat-earth rebuttal, I decided to perform a numerical analysis of the flat-earth model compared to the traditional spherical-earth model.
Science is about testing hypotheses, so let us set up an experiment to test these alternative views. For this, I enlisted the help of two of my children, ages 13 and 11. It made for a fun homeschool project.
The basic problem is the distance between longitude lines in the flat-earth model. In a spherical earth, the longitude lines start from one pole and converge on the other pole, and the distances get wider the closer they are to the equator. But in the northern-hemisphere–biased flat-earth model, the longitude lines start from the north pole and radiate outwards to the supposed ‘encircling southern polar ring’ like spokes on a wheel.
This means that the distances become greater the further south they go. This suggests an easy test of the two models: compare distances to travel times for distant places in the southern hemisphere.
The goal of this simple study was to compare reported airline flight time data with two distance measures, the ‘great circle’ distance of spherical-earth theory and the ‘straight-line’ distance of flat-earth theory.
The driving hypothesis is that the flat-earth map is distorted and so the flight times will not match the calculated distances. This, of course, assumes there is no great conspiracy among the millions of people working for the airline industry or the tens of millions of people who fly on their airplanes annually, which seems reasonable. I am also assuming airplanes on a flat earth would use the rule we all learned in geometry that ‘the shortest distance between any two points on a [standard, Euclidean] plane is a straight line’.
This assumption was in favor of the flat earth model for, as we will see, any curved line would only exacerbate the noticeable distortion with increasing flight time and distance from the source.

First, I obtained a map from the Flat Earth Society website (figure 1). With the help of one of my daughters, we located a suitable city that had many non-stop flights to different cities on other continents. In this case, we chose Johannesburg, South Africa.
We then looked up flights from this city and chose representative destinations on every continent (table 1). After carefully marking the city locations on a printed copy of the map, we painstakingly measured the distances between the cities using a ruler, to the nearest millimeter.
Table 1: Flight data from Johannesburg. Flight times and (great circle) distances were obtained from various online databases and calculators. Physical distances were measured on a globe using a taut string (to the nearest half-cm) and on the flat-earth map using a ruler (to the nearest mm). The globe was approximately 42 cm in diameter. The map was printed to the extents of an 8 ½ x 11 in. sheet of paper. Negative latitudes are south of the equator. Negative longitudes are west of the Prime Meridian.
City | Airport Code | Latitude | Longitude | Flight Time (hrs) | Distance (km) | Measured Distances | |
---|---|---|---|---|---|---|---|
String (cm) | Flat (cm) | ||||||
Johannesburg | JNB | -26.134 | 28.242 | -- | -- | -- | -- |
Antananarivo | TNR | -18.797 | 47.479 | 3.2 | 2,156 | 6.5 | 3.2 |
Mahé | SEZ | -4.674 | 55.522 | 4.9 | 3,752 | 12.0 | 4.2 |
Dubai | DXB | 25.253 | 55.364 | 8.1 | 6,389 | 20.5 | 4.9 |
Dakar | DKR | 14.740 | -17.490 | 8.5 | 6,705 | 21.5 | 5.6 |
Tel Aviv | TLV | 32.011 | 34.887 | 9.1 | 6,473 | 20.5 | 4.4 |
Perth | PER | -31.940 | 115.967 | 9.2 | 8,326 | 26.0 | 12.4 |
Istanbul | IST | 40.976 | 28.814 | 9.5 | 7,430 | 24.5 | 5.0 |
São Paulo | GRU | -23.382 | -46.469 | 9.8 | 7,451 | 23.5 | 9.9 |
Madrid | MAD | 40.494 | -3.567 | 10.3 | 8,077 | 26.0 | 5.7 |
Singapore | SIN | 1.359 | 103.989 | 10.4 | 8,661 | 27.5 | 9.7 |
Frankfurt | FRA | 50.033 | 8.571 | 10.8 | 8,658 | 27.5 | 5.9 |
Zurich | ZRH | 47.465 | 8.549 | 10.8 | 8,387 | 27.5 | 5.6 |
Amsterdam | AMS | 52.309 | 4.764 | 11.1 | 8,986 | 29.0 | 6.1 |
London | LGW | 51.148 | -0.190 | 11.3 | 9,004 | 29.0 | 6.2 |
Sydney | SYD | -33.946 | 151.177 | 11.8 | 11,044 | 35.0 | 15.8 |
Hong Kong | HKG | 22.309 | 113.915 | 12.8 | 10,672 | 34.5 | 9.8 |
Beijing | PEK | 40.067 | 116.600 | 14.1 | 11,699 | 37.5 | 9.5 |
New York | JFK | 40.640 | -73.779 | 15.8 | 12,824 | 41.0 | 9.9 |
Atlanta | ATL | 33.637 | -84.428 | 16.4 | 13,581 | 44.0 | 10.6 |
Next, with the help of my son, we went to the local library because we knew they had a medium-sized globe (approximately 42 cm diameter). We could have simply used the great-circle calculations from table 1, but I felt it was more fair to use my own measurements since I was manually measuring the flat-earth distances. Plus the more error we added, the more difficult it would be to conclude one method was less accurate than the other. It was highly likely that my rudimentary string-on-a-globe method would be wildly inaccurate.
Taking a string, he held the end on Johannesburg while I attempted to make a great-circle to each city. Together, we measured the length of the string from the end to the point where I had pinched it, to the nearest half-cm. This level of precision was chosen so that it did not exceed that of the flat-earth measurements. I wanted to make sure there was a higher level of precision in the theory I do not support. I did not expect much success, and was surprised after I plotted the distances obtained from my string method vs. the great circle distances in table 1 and obtained a line of best fit with an R2 (correlation coefficient) greater than 0.99 (figure 2). According to the string lengths, the scale of the globe was about 313 km/cm.

The final step was to compare the spherical-earth and flat-earth distances with flight times. First, I graphed the great circle flight distances vs. the flight times I obtained from the internet. Given that we are dealing with different airlines, airplane models, airport approaches, and taxiing times, I expected to see more variability in the data. However, a straight-line approximation was obtained, with an R2 greater than 0.97 (figure 3).
This set a high standard for the flat-earth model to meet. The average speed (the slope of the line of best fit) was 871 km/hr, which is a good approximation of the average speed of a modern airliner. The modern long-distance commercial airplane is designed to cruise efficiently at around mach 0.8, with speeds depending on altitude, airplane type, and the speed of local air traffic. Thus, the reported great circle distances are an excellent estimation of the expected flight time. This is another high bar for the flat-earth theory to hurdle.
The next step was to add a comparison of flight speed vs. distance in the flat-earth model, but the data are not in the same units. In fact I had three different units with which to contend: km/hr, cm/hr on the globe, and cm/hr on the flat map. Yet, the solution to this dilemma is simple. The measured distances had to be normalized by dividing by the shortest distance in each dataset.

The normalized data revealed a tight correlation between the spherical earth model and flight time, but no apparent correlation was seen between the flat-earth model and flight time (figure 4). The flat-earth data are so bad that no additional statistical measures were warranted. There is simply no correlation, and the data points that are closest to being true are also the ones closest to the same longitude (i.e. the distances are not distorted by the flattening of the truly spherical earth onto the flat projection).
One might think that, since the normalized flat-earth data are generally below the others in figure 4, that the distances between these cities are shorter in the flat-earth model. However, the normalization process is entirely dependent on the shortest measured distance (in this case, Johannesburg to Antananarivo) and any distortion within the map projection will affect all data points. Since Madagascar is due east of southern Africa, and since both are at a relatively high latitude, the flat-earth projection artificially inflates the true distance.
Comparing the normalized flat-earth distance data to the city list in table 1 demonstrates that the cites with the greatest deviation from expected are the ones that are the farthest east and west of Johannesburg, namely Sydney, Perth, São Paulo, Singapore, and Hong Kong, in decreasing order.
Also, for any group of cities that are more or less along the same straight line from Johannesburg, the closer cities display less error than the farther ones. This is yet another example of the unreality of the flat-earth projection.
One additional measure is the string-length/flat-earth distance ratio. This will tell you proportionally how far apart the two distance measures are and it varies from 2.03 (Antananarivo) to 4.91 (Zurich). In this case, the ratio is highest for the cities in Europe and other places directly above southern Africa on the map, and lowest for cities farthest to the left and right. In fact, the angles of the lines in figure 1 are a direct estimate of this ratio. In other words, the flat map is distorting the true distances in a predictable manner.
Looking out the window
Unless we think all the flight times are wrong and the thousands of passengers are also lying about them, planes would have to be supersonic and have much greater fuel capacity than they do to make many of these routes. Also, the geography of the great circle routes on a global earth is very different from that of the straight-line routes on a flat disk. Passengers on multiple different flights would report seeing very different things out the window if the earth was flat.
For example, passengers on the Johannesburg to Sydney flight have fairly monotonous scenery for nearly the entire trip: the southern Indian Ocean. On a flat earth, the flight would go over the Himalayas (which would be hard to miss!) and other land.

Conclusions
I have set up a simple experiment to test two alternative models of reality. The spherical-earth model came through unscathed. The flat-earth model simply fails to reflect reality. I even made simplifying assumptions in favor of the flat-earth.
Anyone reading this has but two choices: either accept this experimental evidence or reject the reported flight times. However, if you are toying with the latter, please consider that these are backed up by the personal experiences of millions of people and the economic needs of billion-dollar companies in a cut-throat industry where the smallest profit margin might make or break a company.
If you conclude that the flight time data are inaccurate (and they have to be very much so), you are rejecting the type of data upon which your entire society is based. You could not use a computer, drive a car, or read this internet article if this logic upon which this study is based is not real.
If you still want to retreat into the flat-earth position, you are neither siding with the near-unanimous conclusions of 2,000 years of Christian scholarship, the biblical text when properly placed in its grammatical and historical perspective, abundant and simple experimental science, or logic.
Thus, you have not a leg to stand on. Please consider your answer carefully! Also consider that I have not just thrown up a list of facts that supposedly support my own pet theory. I took a lot of time to actually analyze real data. Have you ever seen a flat-earth document that even attempted this level of analysis? No, I didn’t think so.
Readers’ comments
I again rest my case.
How do react when you see thing like this? The person is convinced of his opinion, seems to have data to back up his claims, speaks authoritatively, and throws around mathematical formulae like he knows what he is doing. Seriously, how do you react when you see something like this? Presentations like this are intimidating, but take heart! It usually does not take much digging to solve the riddle, as is true in this case.
For the record, I was referring to his claim that "Atmospheric refraction increases the effective radius of the allegedly spheroidal earth by approximately 4/3 (the k-factor)" when I said 'his math was atrocious'. I should have been more clear and did not mean to mislead anyone or to cast aspersions on his grasp of trigonometry. Since I made that small mistake, I felt I should take some time to answer his mathematical treatise. I did not have to do so and probably will not respond to anything else he posts, but here goes nothing:
First, I got on GoogleEarth and looked up his locations. Indeed, there is about 38 miles (62 km) between these points and the elevations are about what he claimed. I am going to assume he did his calculations correctly but I know that over 40 miles there is enough curvature to hide one location from the other. So, is he right? Should it be impossible to see Souris, PE from Murphy's Pond, NS?
First, note that nearly all such claims involve looking across water. This phenomenon has been known for centuries and atmospheric refraction is the culprit. Ignore his "4/3" rule, for it is certainly being misapplied. All it would take to validate his claims would be to take sightings of various landforms from multiple places simultaneously and apply some triangulation. If you can see point A from point B, how do things look from point C, or point D?
But you don't even have to do that (although it would be fun). Just search the Internet for "Crocker Land". There are two historical examples. The first was from the early 1800s that caused a ship to return from searching for the Northwest Passage when the captain saw high mountains (subsequently named Crocker Land) blocking his path in the distance. Problem is, there are no mountains in that direction (from a point west of NW Greenland). The second example is from another Arctic expedition about a hundred years later that tried to reach Crocker Land, which was easily and clearly seen by them. They traveled across the ice for more than 100 miles, and were forced to conclude that they were indeed seeing something that was not there. They were probably seeing a fata morgana (a type of mirage) created by a combination of the cold but crystal clear air and the extensive sea ice that stretched for miles beyond the horizon.
There are many, many examples of such things from history, from all over the world. But when they are seen, they DO NOT FIT the flat earth model. First, the size of the image is not commensurate with the distance. In fact, the image is the size it should be if the object were much closer that it really is. It is as if you are seeing the object from a closer vantage point. Second, the images are illusory. They can waver, flip over, disappear, reappear, all in the matter of a few minutes. Third, they are not visible under all clear-sight conditions. In fact, it takes a specific set of atmospheric conditions to produce them. Thus, very often you cannot see them even on the clearest days, which should not be true if the earth was indeed flat!
See the image of Toronto in our main article. Notice that you cannot see the base of the tall buildings because they are hidden by the curve of the earth. The same is true of many mirages. As I said above, you are seeing an image that appears closer than it should. Why? Because the light rays bend as they travel through air layers of different temperatures. This causes the light that should be striking in one place to instead strike in another. In cases like this, you are seeing the image as it would be if seen from a closer vantage point.
I must thank Edwin for the comments. He caused lots of discussion here in the office and gave me a chance to read up on some fascinating science, some of which I knew, some of which I had forgotten, and some of which I learned while writing out this answer. If nothing else, his attempted rebuttal has given me (and hopefully you) a better ability to decisively answer the flat earth belief.
However, so what? As a scientist, I could never "rest my case" with one bit of 'evidence' like this. There are always apparent contradictions in science and all scientific positions must wrestle with contravening data. In this article, I presented a conclusive rebuttal of the flat earth, and we presented more details in the original article, but he wants to ignore this information because he 'can see stuff below the horizon'.
Here's how this works: Position 1 (flat earth) attempts to explain the line-of-sight anomaly by saying the earth is flat, but they must reject the published flight schedules of the cutthroat, billion-dollar airline industry and the testimony of tens of thousands of airline passengers and employees. Position 2 (spherical earth) can explain BOTH the line-of-sight anomaly and the airline flight data. The testimonies of the thousands of people involved only confirm the model. Who do you think wins in this case?
But I don't even have to verify the data. If the data the airlines publish are not accurate, they would have gone bankrupt a long time ago. Some in your community want to say these flights do not actually exist. One of the recent commentators even said he 'wanted to meet that person' who had taken the flight in question, yet here we are. It would be impossible to create a conspiracy of this scale and nature, and it would be impossible to maintain it in the face of so many confirming witnesses.
You claim that the flatness can be seen observationally? But we know the atmospheric conditions that allow for seeing things that would not otherwise be along the line of sight. It is called refraction. I would encourage you to look it up, but you obviously either have not read the article(s) or are unwilling to actually engage your brain. Both sides claim to explain this phenomenon, thus it is in no way proof of one side over the other. You need to read How to think, not what to think.
The weight of evidence is so strongly against you that you actually have no theory. All you can claim is "conspiracy" and "people are lying" and things like that. And you claim I do not understand science! Please, pop your head out of "the rabbit hole of deception" and take a look around.
The 8th and 10th men to walk on the moon, Jim Irwin and Charlie Duke, later became Genesis-believing Christians. If any reader is looking for a rocket scientist with a deeper commitment to Genesis, see our interview with Dr. Henry Richter.
As always...Thank you, CMI.
I also have flown from Bali to LA! The great circle route takes you about half-way between Hawaii and the Aleutian Islands. Problem is, my flight went through Hong Kong and THAT great circle takes you very close to Alaska. You are making a non-argument.
For Scriptural references, what do you do with the very same Hebrew word saying the Psalmist will not be moved? See this article for more explanation.
[link deleted per feedback rules]
You say you "tried" to book a flight. Did you really? Or are you just parroting what you have heard from other flat-earthers? Note also that due to the fact that most airlines have major hubs, not all carriers make the direct flight, nor are they expected to. Yet, maybe you should have tried flight QF63/QF64. That is a direct, non-stop, great-circle route that takes you right where it should if the earth were a globe. Here's a generic review (you'll have to ignore the statements about polar bears and chemtrails. It's obviously not written by a scientist!):
http://www.gotravelyourway.com/2013/07/20/qantas-flight-qf63-from-sydney-to-johannesburg/
Note to others: I easily found three CMI speakers that have made this exact and/or similar flights from Johannesburg to Australian cities. Flat-earthers attempt to claim that these flights do not actually exist. I can be adamant about this: they are not telling the truth.
@CT crowd: CMI is doing you a service w/ facts & truth. You listened to them before. Listen to them now!
After consulting with Jonathan Sarfati, we came up with this:
The time zone argument is certainly conclusive against the old flat earth model. Medieval scholars like John Sacrobosco used it to good effect, and it’s much more powerful now (see A flat earth, and other nonsense: Time zones).
But since they claim to explain time zones, one should combine the time zone argument with something else, like the south celestial pole argument (see nearby comment) to make a solid proof.
But note that the modern flat-earth model claims to explain (away!) time zones by postulating that the sun really circles over the proposed disk of the earth. This is so absurdly contrary to what you can see every dawn and dusk: the sun, rises and sets piecemeal below the horizon, and without changing its diameter—it doesn't look like an expanding and shrinking circle as it would if were circling towards and away from us. Also note that the flat-earth model has to avoid even a pretense of being biblically based, because the Bible talks about the sun rising and setting (e.g. Psalm 113:3). As phenomenological or reference-frame language, this is compatible with the global geokinetic view, but certainly not with a sun perpetually overhead.
There one final, and huge, problem with the "spotlight" sun model of the Flat Earth Society: the spotlight would have to light up the earth in a wedge-shaped manner. A round spotlight would not flood everything along the same line of longitude at the same time. Central Africa would be lit before northern or southern Africa as that circular spot of light moved across the earth's surface.
In reality the time zone argument does not work for them, but you have to use the argument correctly.
See
Do I have to believe in a literal creation to be a Christian
and
Can Christians believe evolution
It would be good for you to know what you are arguing against.
And as far as funding is concerned, you cannot call our budget 'significant'. This is even more clear when you consider the enormous funding that our opponents receive. We do a lot with a little, however, so we feel that the donations we receive are not wasted.
Some ancient worldviews did teach the earth was flat with a solid dome over it. Since the Bible is ancient, it should not be compared to modern worldviews in this regards. At worst, the biblical language is equivocal, meaning it does not expressly teach one way or the other. One can take certain passages out of context and attempt to support a flat earth, but then one must ignore other passages that contradict the idea. Yet, the spherical-earth passages seem to trump the flat-earth passages, by which I mean if you take Set A as true, Set B cannot be true (contradiction), but if you take Set B as true, Set A can be true (nested meaning).
I would not put your list of examples in that order, but yes, conspiracy theory is absolutely influencing this trend. The problem here is that they go too far and end up rejecting the Bible, 2,000 years of Christian scholarship, and simple observational science. In the end, their world is a world of magic and deception, which is very much unlike the way the Bible describes God's creation.
You say you and many others have seen distant landmasses? Yes, but there is a limit to the distance you can see and you can only see them under specific circumstances. The entire phenomenon (mirage) is explained by the diffraction of light by air of different temperatures/densities. This is simple, laboratory science. So, since both your side and our side expects to see this, it is a non-experiment. Thus you cannot use it to support your side! Worse, I believe there are certain aspects of that phenomenon that contradict your model, but there is no room to explain that here.
You say the Devil influences science? I can accept that so far as he influences people. But the Devil did not create this world. He did not create science. He did not create you. And he did not give you your ability to reason, to observe, or to draw conclusions. He is not in control. Thus, we are at liberty to use our God-given faculties for reason to draw conclusions about the world we live in. And when we do so, the earth is clearly a sphere. To say otherwise is to deny reality. Come away, my friend.
It is so sad to see so many fellow-believers following after this flat-earth nonsense. Friends, do not give the atheists a legitimate reason to mock believers!
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