Estimating distance to horizon

This problem can be solved by assuming that the earth is a perfect sphere. In fact, if we consider a slice through the centre of the earth, we can treat this problem in two dimensions and therefore consider just a circle. The problem then reduces to finding d in the right angle shown in the figure:

where h is the height of the observer's eyes and R is the radius of the circle (or the earth in our case).

Pythagoras' theorem tells us that we can relate the quantities (R+h), R, and d (the three sides of the triangle in the figure) by:

R^2 + d^2 = (R+h)^2

which we can solve for d. (Note: ^ denotes "raised to power of".)

(Of course, having looked at the diagram, we should realize that the distance we actually want is represented by an arc but the line segment corresponding to d should be close enough for our purposes, especially as the relative scales of R and h have been grossly exaggerated!)