So we can make the following points:
The rest of it is some geometry, and knowledge about solid angles (or surface of a sphere), which is not really interesting and hard to demonstrate without boring figures. But, overall, the formula comes out to be extremely simple. The fraction of (one-half of) the surface of earth one can see at an altitude h is simply given by:
F = h / (h + R)
where R is the radius of the earth. So, if you go about one earth radius high (which is about 6370 kilometers or 3960 miles) you can see 1/2 of the most you can see. Once you go as far as the moon (roughly 400 000 kilometers away) you can see about 98.4% of one face.
Once again, the answer really depends on the exact definition of the question. If 90% is good enough for you, you only need to go 57000 kilometers (35600 miles) high from the surface of the earth. If you want 100%, that you just are not going to get at any finite distance.
Answered by: Yasar Safkan, Ph.D., Instructor, Yeditepe University, Istanbul, Turkey
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'I believe there is no philosophical high-road in science, with epistemological signposts. No, we are in a jungle and find our way by trial and error, building our road behind us as we proceed.'