If a bird fly to height of $h$ ,What's the area that it can see? Suppose a bird fly to height $h$ from earth . The bird can see area under by it's eyes ,name as $S$ ,What's $max \{S\}$ ?
Is it possible to solve  ?
my first trial was to assume a cone by $height =h$ and $S=\pi R^2$ as area like the figure I attached with.
Is $S$ a constant for special amount of $h$ ?
can we calculate $max \{S\}$ or $S$?(or something is missed  ?)
Thanks for any hint ,in advance.
 A: I am assuming that there is no limitation on the angle of vision of the bird.
The only known values I am assuming is the half apex angle $\theta$.
By property of tangent, the half angle subtended at the center will be $90-\theta$.
By using the solid angle formula,
$$
A=\phi r^2=2\pi(1-\cos{(90-\theta}) = 2\pi (1-\sin{\theta})
$$
A: Assume the earth to be a sphere of radius $r$ with centre $O$. Let the bird be at a point $B$, with a point $H$ on the horizon as seen by the bird. Then $OH=r$, $OB=r+h$, $HB\perp HO$, and $$\sin\angle OBH=\frac r{r+h}.$$The area of the spherical cap seen by the bird is$$2\pi r^2(1-\cos\angle BOH)=2\pi r^2(1-\sin\angle OBH)$$$$\qquad=2\pi\frac{r^2h}{r+h}.$$
A: I became interested in this problem and the word "bird" caught my attention.
We have a geometric approach given by John Bentin.
But birds' eyes have a lateral, a frontal and a blind areas of vision in the horizontal plane.
I couldn`t find anything in the vertical plane.
See http://people.eku.edu/ritchisong/birdbrain2.html
Besides there are three kinds of visual fields
http://estebanfj.bio.purdue.edu/birdvision/visualfields.html
Here https://en.wikipedia.org/wiki/Field_of_view
we can find two sketches showing the horizontal and the vertical fields of view.
Also found that there is a visual-field projections on orthographically viewed spheres (that I am trying to understand)
https://www.researchgate.net/publication/313038494_White-headed_Vulture_Trigonoceps_occipitalis_shows_visual_field_characteristics_of_hunting_raptors
and 
https://www.researchgate.net/publication/232225655_Extreme_binocular_vision_and_a_straight_bill_facilitate_tool_use_in_New_Caledonian_crows
