>In spherical geometry, the interior angles of a triangle add up to more than π. And in fact you can determine the area of a spherical triangle by how much the angle sum exceeds π. On a sphere of radius 1, the area equals the triangle excess

To all the flat earthers out there, this property can be used to find out earth is not flat, just by drawing a giant triangle on the surface, without leaving the earth. Historically, to prove the earth is round, people have relied on the sun shining directly overhead on wells in different cities. But this approach proves it without the need to refer the sun.

Once you internalize that flat-Earther-ism isn’t about the Earth being flat you realize that rational arguments are pointless.

To expand on that, it’s about community and finding people who share your interests. The movie Behind The Curve explores this idea and it’s quite revealing.

> relied on the sun shining directly overhead on wells in different cities.

It was just one city actually. The critical piece is that the city's northern latitude was nearly identical to the Earth's angle of axial tilt. Which also means that this shadow phenomenon only occurs during the Summer Solstice.

https://www.khanacademy.org/science/shs-physical-science/x04...

>Historically, to prove the earth is round, people have relied on the sun shining directly overhead on wells in different cities.

That wasn't to prove the Earth is round (and it doesn't prove it). Eratosthenes assumed two things when he performed his experiment: 1) the Earth is round, and 2) the Sun is an infinite distance away. By just this experiment he would have been unable to distinguish between this situation and the Earth being flat while the Sun being only a finite distance overhead (and in fact a fair bit closer than it actually is). Eratosthenes and his contemporaries were already convinced of the roundness of the planet, and he simply wanted to measure it.

>But this approach proves it without the need to refer the sun.

A flat-earther would just tell you that you're not able to maintain a straight path over such long distances without relying on external guides that would definitely put you on curved paths. If the Earth is flat and you stand at 0 N 0 E, how do you move in a straight line East of there? I.e. continuously moving towards the South because the polar coordinates curve towards your left as you progress.

As they say, you can't reason someone out of something they didn't reason themselves into in the first place.

> But this approach proves it without the need to refer the sun.

Only if you're happy "proving" your argument to an audience that never had any doubts. You can't use this argument to prove the earth is not flat over the objections of your audience because you can never convincingly show that any given line is straight.