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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'In a way science is a key to the gates of heaven, and the same key opens the gates of hell, and we do not have any instructions as to which is which gate.
Shall we throw away the key and never have a way to enter the gates of heaven? Or shall we struggle with the problem of which is the best way to use the key?'
Richard Phillips Feynman