# Answer to Riddle #67: How Far is the Horizon

67. How far is the Horizon?

This is not a typical 'Puzzle' but it is some common sense geometry. I saw some puzzles of the form 'John is stood on the beach...' I thought these all come down, basically, to working out the distance to the horizon. So let's just do that.

Before reading the answer can I interest you in a clue?

There are a number of ways to make this calculation, the important thing is that you can justify any assumptions you make.

AOB is a right angel triangle where from O the opposite side 'd' is the distance to the horizon, the adjacent side 'RE' is the radius of the earth and the hypotenuse is 'RE' + 'h' where 'h' is the height of the observer. We will apply Pythagoras Theorem

X2 + Y2 = Z2
RE2 + d2 = (RE + h)2
RE2 + d2 = RE2 + 2REh + h2
d2 = 2REh + h2
d2 = h(2RE + h)
d = ±√h(2RE + h)

The algebra to now should be fairly easy to follow. Note the ± is simply because in our 2D model the horizon is in both directions, not strictly necessary, but good for completeness.

On the right we have ±√h(2RE + h) and we will be approximating this to √2hRE. Some people would argue that the approximation is less accurate, but the earth is not a sphere, it's radius varies by ±15km and so using 2RE + h seems like an accuracy we can't claim.

d = √2hRE

Given RE = 6,371,000m for me at least whose eyes are 6ft2 from the ground or 1.8796m the horizon is 4,894m away.

## Two Objects Above Sea-level

So, to now, we have considered an observer for example stood on a beach looking at the horizon and calculating that distance. What if you have someone on the shore looking for the top of a ship's mast?

Since the limiting factor in how far we can see is the hump of the earth in the way as it just disappears ABC will be a straight line. We can simply add the tow horizon distances together. As in

d = d1 + d2 = √2h1RE + √2h2RE

As you would expect both AI's did well at this.

If you're curious what Bard made of this puzzle...

If you're curious what ChatGPT made of this puzzle...

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