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.

**Answer**

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

r2+d2=(r+h)2.

It follows that

d2=(r+h)2−r2

so

d=√(r+h)2−r2.

This admits simplifiction:

d=√(r+h)2−r2=√(r2+2rh+h2)−r2=√2rh+h2.

When h is tiny compared to r, we can say

d≈√2rh.

**Attribution***Source : Link , Question Author : M.Herzkamp , Answer Author : Michael Hardy*