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.

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$$2\pi r^2(1-\cos\angle BOH)=2\pi r^2(1-\sin\angle OBH)$$$$\qquad=2\pi\frac{r^2h}{r+h}.$$
