# How far can one see over the ocean?

Since Earth is a sphere, one has only a limited visibility radius. How far is that, actually?

This Q&A was inspired by this question, about whether or not Legolas can see the 24km distant Riders of Rohan.

I have up-voted the answer by M.Herzkamp, but I also think he makes it somewhat more complicated than it needs to be. The distance from the center of the earth to your eye is $r+h$, where $r$ is the radius of the earth and $h$ is the height of your eye above the ground. The distance from the center of the earth to a point on the horizon is $r$. The distance from your eye to the point on the horizon let us call $d$. The three sides of a right triangle are then the legs, $r$ and $d$, and the hypotenuse $r+h$. Applying the Pythagorean theorem, we have
When $h$ is tiny compared to $r$, we can say