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How far would I have to go to see a fully rounded Earth?

Recently, I saw a video on Youtube in which a sky diver called Felix Baumgartner ascends to $120,000$ feet (= $39$ miles) in a stratospheric balloon and make a freefall jump, rushing toward earth at supersonic speeds.
I could see the full face of the Earth in this video from a height of approximately 39 miles. This prompts me to ask this question.

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    $\begingroup$ When do you consider an object to be seen round? I guess you would have to consider the eye resolution. $\endgroup$
    – jinawee
    Feb 23, 2014 at 11:59
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    $\begingroup$ Two comments: 1) the jump altitude is 39 km, not 39 miles; 2) the wide angle head-camera used for recording Baumgartner's jump creates a 'barrel distortion' of the images, causing an apparent spherical horizon; in reality the horizon looks pretty straight at altitudes of 39 km (an altitude corresponding to 0.1 % of earth's circumference). $\endgroup$
    – Johannes
    Feb 23, 2014 at 15:29
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    $\begingroup$ Seems to be a duplicate of physics.stackexchange.com/q/25509 $\endgroup$ Feb 23, 2014 at 16:06
  • $\begingroup$ @Johannes is explaining what Neil deGrasse Tyson claimed without proof: twitter.com/neiltyson/status/431554750119550976 $\endgroup$
    – user10851
    Feb 24, 2014 at 3:58

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This will be limited by our field of view (FOV). I couldn't find a better source, but Wikipedia says the vertical range of the field of view in humans is typically around 135° and the horizontal range around 180°.
enter image description here

So for the Earth to be entirely within your field of view it will be limited by your vertical range. And by using a little bit of geometry you can find the height above the surface of the Earth, $h$, if you approximate it as a perfect sphere with radius $R$=6371.0 km: $$ h=R\left(\frac{1}{\sin\frac{\theta}{2}}-1\right) $$ which gives: $h\approx$ 524.9 km (=326.2 miles).

However if you would look trough a lens, which would increase your FOV, this height could be significantly be decreased. The camera used to capture the sky diver will probably have had a higher FOV. The minimum FOV at a height of 39 miles (=62.76km) would have been 164°.

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  • $\begingroup$ That was a nice analysis.FOV matters here $\endgroup$ Feb 24, 2014 at 9:22
  • $\begingroup$ So technically, with the right lens, in the middle of the ocean, the answer would be "a few dozen meters or so"? $\endgroup$
    – user371366
    May 19, 2017 at 6:10
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    $\begingroup$ @dn3s Yes. $\endgroup$
    – fibonatic
    May 19, 2017 at 6:16
  • $\begingroup$ @fibonatic huh, that's much better than my far-fetched ocean scenario $\endgroup$
    – user371366
    May 19, 2017 at 6:19
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Here's a contrarian opinion: there's no such thing as seeing a fully rounded earth!

Standing at the top of a high building, you look out from the center of a circular (fully round?) disc (with a hump from the spherical earth). The edge of that disc is the horizon, the farthest point you can see in any direction.

Since you are above that rim, you are looking down very slightly. Navigators using a sextant have to account for this "dip of the horizon" when making sextant observations of the sun or stars. The height of the navigation bridge above the water is a correcting factor in their calculations.

As your altitude increases, two things happen. The distance to the horizon increases, but more slowly as you rise. Secondly, you are looking farther and farther down to see this (fully rounded?) disc, as you rise farther and farther above it.

B as you rise you are always looking at a round disc, from a slowly changing perspective...

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