A little something to play with, as this conversation comes around on a regular basis. Basically two questions, on a globe what would be the distance to the horizon when up a tower, and what angle would that be at, down from the horizontal? The issue with answering such things is the comprehension of scale - earth big, us and even the biggest towers are comparatively small.
In this simulation (you can zoom in and out):
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you can change the radius of the planet and a tower height up to 10km. I've picked 600m as the starting point for a tower as that is inline with the worlds highest sky scrapers. From there the distance to the horizon is reported as 86 km and an angular drop of 0.7 degrees. This site agrees working it out mathematically:
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It gives 87.5km. The difference is the granularity of the sliders and it is not exactly 0.6km underneath.
Pull the slider down to the smallest it will register (0.1km - I could not get more than one decimal place) that is a 100m tower and an angle to the horizon of 0.2 degrees. To actually see the tower you have to zoom right in (middle scroll wheel) and at that point the curve of the earth appears in the diagram as being so slight it almost looks like a level line. Pushing the height of the tower up to 5km high and you have a drop of 2.2 degrees. All well within an expected field of view of the human eye and most optical instruments. If you wanted, you can also change the size of the planet. (Earth has a radius of 6731km)
What it shows is just how huge the earth is, in comparison to us and all this falling backwards unable to see the ground is a load of bollocks. We have tended to show diagrams where the height of a tower or observer is massive, perhaps a 5th of the planet radius. That is over 1000km high so yes on that sort of scale you would be looking down a long way to see the horizon. I can see why flat earth conspiracy nuts get confused, but it is like often said on this thread, draw it out to scale and it makes a lot more sense.
Supplied because I thought many on this thread will find it interesting. I'm not expecting it to be used to convince a certain somebody, who will say 'but but, nah bollocks, deflection, look squirrel'.