It seems like most people disregard conspiracy theorists who think NASA is a complete hoax and somehow the planet is flat, believing they’re just trolls. Perhaps a lot of people who argue in favor of a flat earth are simply playing devil’s advocate for the fun of it, but there are still some who definitely believe it, many for religious reasons. I think the combination of math, science, religion, and stupidity makes this a perfect subject for my blog.

One of my favorite new YouTubers recently released a video on this topic, and I thought it would be fun to add something to the conversation. Rather than deal with the ridiculous arguments and complete disconnection from reality that comprise the flat earth delusion, I’m going to attempt to provide a simple representation of the mathematics involved. I could cite the direct airplane flights from South Africa to Australia, which are significantly shorter than they would be on a flat earth. Or the ancient Greeks who calculated the circumference of the earth with remarkable precision many hundreds of years before Christianity began. Or that time I watched the ISS with my own eyes as it passed over me in orbit. Or a huge number of other things. But perhaps what some people really need is a simple math lesson.

Therefore, I have decided that I will calculate the circumference of the planet using a silent 20-second YouTube video.

I stumbled across this video a while ago. It shows the top of the Duke Energy Center in Charlotte, NC, viewed with an extreme zoom from a place called World’s Edge in the mountains of western North Carolina. When I first saw the video I left a detailed comment containing the simple mathematics that, in conjunction with the footage, prove the earth is curved. Perhaps that’s one of the reasons the comments have since been disabled. This article is essentially an improved reproduction of my original comment.

In order to analyze the significance of what is seen in the video, I need to gather some basic information about the distances and elevations involved. The differences between the elevations will allow me to draw triangles, do some basic trigonometry, and discover what we should expect to see if the earth were flat. The same data will also allow me to calculate the circumference of the earth based on what we actually see in the video.

The elevation at World’s Edge is 803 meters. Charlotte is at 229 meters, and using this handy tool I can see that the highest elevation between them is 331 meters. The distance between Charlotte and World’s Edge is 129 kilometers.

I couldn’t scale down the measurements and keep the proportions of the triangles, because at 12.9 inches for the distance, the elevation difference would only be 0.057 inches. A triangle that thin wouldn’t do much good for an illustration. To make it work, I had to exaggerate the vertical axis by a factor of ten. In order to accurately represent the height of the ridge, I calculated that the difference between the elevation of the ridge and Charlotte is about 17.7% of the difference between World’s Edge and Charlotte. Since the vertical height of my illustration is 55 pixels, and 17.7% of that is 9.735, I made the ridge 10 pixels tall.

But wait, you might say, what if that ridge is really close to Charlotte? Couldn’t it block your view then? And you would have a good point, which is why we still need to do a little trigonometry. Fortunately, there’s an internet tool for that as well. Our first triangle is one with a hypotenuse that stretches from World’s Edge to Charlotte, and I’ll call it WCX for short. The distance we want to find is the longer leg of a right triangle that shares its small angle with WCX and has a height of 102 meters (331 minus 229). A ridge 331 meters in elevation would have to be within that distance from Charlotte in order to obscure the city from the view of World’s Edge…if the earth was flat.

We know the legs of WCX are 574 meters (803 minus 229) and 129,000 meters (129 kilometers). Plug those numbers into the triangle calculator and it tells us the small angle is 0.255 degrees. Input that, along with a height of 102 meters, and our solution is 22.9 kilometers. Looking along the straight line between Charlotte and World’s Edge, I found a city called Gastonia and measured its distance from Charlotte at 32 kilometers. Nowhere between the cities does the elevation rise above about 250 meters, and after further analysis of the terrain it’s clear that there are no ridges tall enough and close enough to Charlotte to block it from the view of World’s Edge on a flat earth. You would be able to see the entire city.

What we actually see in the video is a small portion of the top of the tower peeking out from behind a ridge. Based on pictures of the tower, I’ve concluded that the angled part at the top is about 25% of its overall height. In the video, this is the only part of the tower that is visible, so the line of sight from World’s Edge to Charlotte is reaching the 240-meter tower only 60 meters below the top. That means there is a ridge somewhere between the camera and the tower which is somehow obscuring not only the entire city of Charlotte, but also an additional 180 meters of elevation, despite being under 331 meters and/or more than 22.9 kilometers from Charlotte. I made a little tweak to my previous illustration to show a phenomenon that could be responsible for this apparent mathematical impossibility.

But does the observed discrepancy match up to the known circumference of the earth? Let’s break out the math again.

To make this simpler, I’m going to assume the ridge hiding Charlotte from view is the one in Gastonia, 32 kilometers from Charlotte and 250 meters in elevation. From my research I believe this is a correct assumption, since I can’t find anything else tall enough and/or close enough to Charlotte to appear higher than the Gastonia ridge. So a straight line of sight runs from 803 meters, to a 250 meter ridge, and then hits the tower at an elevation of 409 meters (229 plus 180). How does the ridge appear in line with something 159 meters higher than it? Curvature.

32 is almost exactly 25% of 129. So we need to find the full height of an arc that is 129,000 meters long, on which a point at the one-quarter mark is 159 meters high. The height and length of this arc will tell us the radius of the full circle it belongs to, after we plug them into a handy online circle calculator. Since we’re dealing with a very shallow arc, I can probably get away with assuming that the height of the one-quarter point will be roughly half of the arc’s full height in the middle. After I plug in a length of 129,000 and a height of 318, the circle calculator tells me that the radius of the planet is 6,541 kilometers. However, this is the circle at an elevation of 229 meters, which I’ve been using as a baseline for my trigonometry, and I want to know the circumference at sea level. So I subtract 229 for a radius of 6,312, which yields a circumference of 39,639 kilometers.

According to Space.com, the circumference of the earth at the equator is 40,030 kilometers. My margin of error is 0.9%, which I’d say is pretty good considering all the rounding I did, the rough distance measurements, and my imprecise method of finding the height of the arc. In contrast to this, the figures cited in the video description claim that over 129 kilometers, the ground would “drop” 1300 meters (4,266 feet) and therefore the tower would be far below the horizon. That is completely wrong; an arc with a length of 129,000 meters and a height of 1300 meters would give you a planet with a circumference of only 10,045 kilometers, which is less than the circumference of the moon.

I made an illustration to show that the mathematical mistake made by every flat-earth believer gives them a roughly four-fold exaggeration of the actual curvature, which means the spherical earth they are debunking is a straw-man planet four times smaller than the real earth.

So now you know how to prove the earth is spherical *and *calculate its circumference within one percent accuracy using a YouTube video, a few elevation and distance measurements, and math. The funniest part, which makes this whole thing so ironic, is that the video seems to have been intended as “proof” the earth is flat. They shot themselves in the foot with a zoom lens.