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Can we calculate the area of sky visible to us from the point we are standing?I mean is there any idea or experiment to calculate it?

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Approximating the Earth as a sphere with radius $R$, then when viewing from a height $h$ above the surface, the Earth blocks out a cone of some opening angle $2\vartheta$, where $\csc\vartheta = 1+\frac{h}{R}$. Thus, the visible portion has a solid angle of $$\Omega = 2\pi\left(1+\cos\vartheta\right) = 2\pi\left(1+\frac{\sqrt{h^2+2Rh}}{R+h}\right)$$ steradians. Divide this by $4\pi$ to obtain the fractional area of the the visible sky, compared to what you could have if the Earth wasn't blocking your view, since a full sphere subtends a solid angle of $4\pi$ steradians. That is probably a more natural measure of the visible sky than a literal area.

For an actual area, you need some sort of reference distance $r$ to measure from, with the visible sky a distance $r$ away having area $A = \Omega r^2$.

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For the area of sky as defined by "areas visible above your head inside Earth's atmosphere" then the answer depends on the height of the objects that you would like to observe. I like to observe airplanes, so without loss of generality I'll use ~10,000 meters as my "sky".

We can see airplanes up to about 40 KM away.

π * (40)^2 ~= 5000 KM^2

The altitude:area relationship is a squared relationship, therefore if you decide that you are interested in only 1000 meters as your "sky" (maybe you like to photograph birds) then you have about 3 KM^2 of sky available to you.

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