How Far Away Is The Horizon?

While pursuing attractive landscape photographs, I have taken many pictures that include the horizon. From time to time, a question occurred to me: how far away is that line I see?

For example, in the photograph below which I took while sailing off the island of Samos, what is the distance between me and the horizon?

At first, this may seem like a question that is almost impossible to answer. With a little thought, however, it turns out to be quite straightforward.

Since this blog attempts to reflect the beauty contained within the area I can see, I decided to calculate that distance—in other words, the radius of my visible world.

And it can be done rather simply.

The moment mathematics enters the discussion, many people immediately turn their eyes elsewhere. Still, with a little patience, you can see how the calculation works.

Consider the graphical masterpiece I produced below. The blue circle represents the Earth. I am standing at sea level, looking towards the horizon.

Let us assume that the Earth is perfectly spherical and that its radius is 6,370 kilometres, approximately its value at the Equator. In other words:

r = 6,370,000 metres

My height is:

h = 1.78 metres

What we want to find is the distance to the horizon, represented by u.

Since the diagram contains a right-angled triangle, we can use the famous theorem of Pythagoras: the sum of the squares of the two shorter sides is equal to the square of the hypotenuse.

In this case, the hypotenuse is the radius of the Earth plus my height.

(r + h)² = u² + r²

Expanding the expression gives:

r² + 2rh + h² = u² + r²

After cancelling the terms that appear on both sides:

2rh + h² = u²

Since we are interested not in the square of u, but in u itself, we take the square root:

u = √(2rh + h²)

When the values of r and h are placed into the formula, the result is:

u = 4,672 metres, or approximately 4.67 kilometres.

So, when standing at sea level on a clear day and looking towards the horizon, the visible limit lies slightly less than five kilometres away.

It was simply something I had wondered about, so I thought I would share it.

Update:

Increasing the height of the observation point above sea level greatly extends the distance we can see.

When I took the photograph below, the aircraft was flying at an altitude of 10,600 metres.

If we disregard the line of clouds obstructing the view and enter 10.6 kilometres as the viewing height in the same formula, the distance to the horizon is approximately 365 kilometres.

So, if you wish to see farther, it genuinely helps to rise a little higher.