exupero's blog

Apparent size above horizon

In an earlier post I estimated the distance to the horizon given the radius of the planet and a height above the ground. In this post, we'll find the apparent height of an object whose base is beyond the horizon. In the diagram below, we know the radius of the planet R, our height above the ground h, the object's height H, and the distance between us and the base of the object s. We want to find θ.


To do so, we can find α from the definition of radians:

γ from the definition of sine:

d from law of cosines:

and β from the law of sines:

Solving these for the needed values and plugging them into θ = β - γ gives us

which has a nice enough symmetry that I won't spoil by inlining d.