World is in fact (99.665%) spherical
A high-school level mathematical proof (for the hard-headed flat-earther).
There is a nice Discovery Science (now called Science Channel apparently) segment showcasing the famous ship disappearing over the horizon effect. I was able to find the archived version from Wayback Machine even if the original was deleted. I recommend watching, but is not necessary.
They use a laser at different distances to project a straight line towards a lake and mark a board on their boat at each respective points.
If, travelling through the lake, they are indeed travelling in a straight line, you would expect to see the laser pointing to same height no matter how far they venture.
If, on the other hand, they are travelling in an arc, ever so slightly bent, you would expect to see the laser pointing more and more upwards the farther you go, as though you are sinking.
It turns out, the laser does actually point upwards! As they go 3 miles away from the shore, the laser beam rises by 6 feet! Good enough? For most people. But you can just assume it’s fake. Sadly, they do not actually use those 2 figures in any way other than to show that the height of laser beam in fact rises.
Therefore, I will prove by way of simple geometry and trigonemetry that this fact alone is enough to estimate the radius of the Earth, and if it does match up to the one we empirically know of, it is a very good reason to believe this experiment was genuine.
Given that the boat has traveled 6 miles (4.828 km) to see the laser beam rise by 6 feet (1.829 m), it would follow that a=4.828 and b=1.829×10⁻³
Arc length traveled can be found by:
Substituting a, we get:
From the small triangle:
Given that b=c for small θ, and substituting the equation of θ above:
Cosine approximation for small angles (r >> 4.83) are:
Plugging θ, you get:
Solving for r, the positive value (since radius is positive!) is:
As you see, this is impressively close to the empirically found average radius of the earth (average since it is slightly different at different points! Remember 0.335% oblateness), which is 6371 km. In fact, it’s only 0.047% off, even though the Discovery Science measurements were pretty crude.
Hopefully this gives some intuition as to how a simple experiment like this can easily lets you derive the gigantic Earth’s radius within 0.05% margin of error, and that this knowledge is incredibly accessible.