Lake Michigan Photos Show Earth’s Curve With Simple Math

Yes, the Earth’s surface curves even over a lake you can see across, but the bend is so gentle that your eyes will almost never catch it from the shore. That is the whole point of a playful photo essay by Greg Pagel, a 47-year-old musician from Manitowoc, Wisconsin, who posted it on Imgur under the name GregPagel in 2018.

Pagel had stared at the Lake Michigan horizon for most of his life and kept wondering whether he was seeing a curve or imagining one. So he photographed the view near his home, measured the stretch of water in Google Earth, worked out what fraction of the planet that stretch represents, and drew it to scale. His answer was an arc of roughly 0.12 degrees: real, but almost invisible.

His timing was good. Flat-Earth jokes were everywhere online that year, partly because of a widely shared exchange between Elon Musk and the Flat Earth Society on Twitter. Pagel’s post went the other way: instead of mocking anyone, it showed, with a calculator and a few diagrams, why a flat-looking horizon is exactly what a round planet should look like from a beach.

Below, each of his images is explained in order, followed by the geometry behind them and other simple observations anyone can use to see that the Earth is round.

The view from the Manitowoc shoreline

The first frame is the scene that started it all: a paved lakeside path, a single young tree and a rocky shore under a bright summer sky. Lake Michigan stretches out to a horizon that looks ruler-straight.

Pagel’s own caption explains that he took the picture a few miles from his house in Manitowoc. A stone breakwater runs out into the water on the right, which becomes a useful landmark later in his measurements.

A red arrow marks the beach at Silver Creek Park in the far distance to the right. This is one of the two points he later uses to measure how much open water his camera actually saw.

The second landmark sits to the left: the south side of Two Rivers, the neighbouring town where Pagel grew up. With one point on each side of the frame, the view now has a measurable width.

Here he pauses for a local joke about why Two Rivers got its name, comparing it with a less flattering description of the shoreline. It is a small detail, but it sets the tone of the whole post: curious, friendly and in no hurry to sneer at anyone.

Why the horizon looks flat from the beach

Now comes the question many people quietly ask themselves at the seaside. Pagel points at the horizon, notes how level it looks, and asks, tongue in cheek, whether he has been lied to about the shape of the Earth.

He zooms in to get a better look. Magnified, the line between water and sky still appears dead level, with no obvious bend from one edge of the picture to the other.

A tighter crop with a single word stamped on it, “FLAT.” This is the honest result of looking: from a standing height, the eye cannot pick out any curvature, and Pagel does not pretend otherwise.

Even a wide panorama, which takes in far more of the lake, gives the same impression. A curved-looking panorama would not prove much anyway, because panorama stitching can bend straight lines on its own.

There is a simple reason for all this. For someone standing on a beach with their eyes about 1.7 metres above the water, the geometric horizon is only about 4.7 km (2.9 miles) away, according to the standard formula used in navigation. Normal atmospheric refraction pushes it a little farther, to around 5 km. What you see is the edge of a small circle centred on you, cut out of a sphere that is about 12,700 km across. At that scale the curve is far below what the eye can detect. The geometric horizon only sinks noticeably below eye level once you climb much higher; from a hill 1,000 metres tall, it sits about a degree below a level line.

Measuring the view with Google Earth

This screenshot shows where Pagel stood, with the Two Rivers shore on one side and Silver Creek Park on the other. Because the shoreline curves around him, the land blocks his view beyond those two points, so the stretch of water between them is everything his camera could see.

Google Earth’s ruler gives the straight-line distance between the two landmarks as about 8.4 miles (13.5 km). He overlays that line on his panorama so readers can connect the map with the photo.

Then the arithmetic, shown in his handwritten notes. Taking the Earth’s circumference as 25,000 miles, he sets up a simple proportion: 8.4 miles is to 25,000 miles as x degrees is to 360 degrees. The result is about 0.12 degrees, and he even checks his answer by running the proportion backwards.

His rounding holds up. The Earth’s circumference is about 40,075 km (24,901 miles) around the equator, so 8.4 miles works out to roughly 0.121 degrees of arc. Strictly speaking, his line is a chord across the bay rather than a path along the surface, but over such a short distance the difference is negligible.

Picturing a 0.12-degree arc

To make the number meaningful, he draws the Earth as a plain blue circle and asks what a 0.12-degree slice of its edge would look like.

He starts big. A 90-degree arc, a quarter of the circle, is impossible to miss and curves dramatically.

Halving it to 45 degrees still gives an obvious bend. The idea is to shrink the slice step by step until it matches the view from the lakeshore.

At 1 degree, the picture changes completely. Zoomed in until the slice fills the frame, the edge of the circle already looks nearly straight, and he asks the reader to find the curve.

He reminds us that his lake view covers barely an eighth of that 1-degree slice. If one degree already looks almost flat, the much smaller arc he actually photographed should look flatter still.

Here is the 0.12-degree arc itself, drawn to scale and stretched across the image. The top edge droops only very slightly towards the sides.

An arrow points at the slight bend he wants us to notice. It is subtle, but it is there, and that is precisely his conclusion: “almost nothing” is not the same as nothing.

Back on the lake: what the photos really show

The final photo returns to the water with the same message: the curve is present across that view, but at 0.12 degrees it is too small to see. A flat-looking horizon is not evidence against a round Earth; it is what a round Earth of this size predicts for anyone standing near the waterline.

One honest caveat makes his point even stronger. The horizon you see from a beach is not really the 8.4-mile line across the bay; it is a circle a few miles away, and the slight sagging of that circle is smaller still. Either way, the lesson is the same: ground-level eyesight is the wrong tool for judging the shape of a planet.

Other simple evidence that the Earth is round

If the horizon cannot settle the question, what can? People worked out the answer long before cameras or satellites.

• Eratosthenes’ shadows. Around 230 BC, Eratosthenes compared the noon Sun at Syene (modern Aswan), where it shone straight down a well on the summer solstice, with Alexandria, where shadows fell at about 7.2 degrees, or one-fiftieth of a circle. Combined with the distance between the cities, that gave a circumference remarkably close to the modern value.
• Lunar eclipses. By about 330 BC, Aristotle was offering observational arguments for a spherical Earth. The best known is the curved shadow it casts on the Moon during every lunar eclipse, whatever direction the Moon is in.
• Things vanish from the bottom up. Distant ships, coastlines and city skylines across wide water lose their lower parts first, because the bulge of the water hides them.
• Different stars in different places. Travel north or south and new constellations rise while others sink, which only works on a curved surface.
• Going all the way around. The Magellan–Elcano expedition of 1519 to 1522 sailed around the globe, and today airline routes, GPS satellites and photos from orbit all depend on the same round geometry.

You can try a version of the first idea at home: friends in two cities several hundred kilometres apart, measuring the shadow of a vertical stick at local noon on the same day, will get different angles. On a flat surface lit by a distant Sun, those angles would match.

Common flat-Earth myths, briefly

“People in the Middle Ages thought the world was flat.” Most medieval scholars knew it was round. Historians trace the myth to 17th- and 19th-century writers, with Washington Irving’s romantic stories about Columbus doing much to spread it.

“The horizon always looks flat, so the Earth must be flat.” As Pagel’s diagrams show, a level-looking horizon is exactly what the geometry predicts at ground level.

“Flat-Earth belief is an ancient tradition.” The modern movement is relatively recent. It traces back to Samuel Rowbotham’s 1849 pamphlet Zetetic Astronomy, and the group known as the Flat Earth Society was founded by Samuel Shenton in England in 1956. Surveys suggest firm believers remain a very small minority.

What makes Pagel’s post worth revisiting is its method. He did not appeal to authority or ridicule anyone; he asked a fair question, measured something, did the sums, checked them and drew a picture. That is a habit anyone can borrow, whatever the topic.

Sources: Photos and diagrams by Greg Pagel (Imgur gallery, YouTube); interview in Bored Panda; Wikipedia: Horizon; Wikipedia: Eratosthenes; Wikipedia: Flat Earth; Wikipedia: Spherical Earth.