How far away is the horizon?

I came across this question during my investigation of the Britam Tower, Nairobi, as part of my A-Z buildings project. (Click here for that.) According to a number of online sources, from it you can see both Mount Kenya and Mount Kilimanjaro, which are more than 100km and 200km away respectively. So I wondered, how far away is the horizon?

When you’re standing looking out at a flat plain, such as desert or sea, there is a fixed distance you can see, until the earth’s curvature makes it impossible to see any further. Turns out this calculation is actually relatively simple. (My own work! Relying on logic.) Here is my simple diagram:

• r represents the radius of the earth
• h represents the height of the observer
• x represents the distance to the horizon – what we are trying to find

We know that angle CAB is a right angle – this comes from the circle theorem stating a tangent (AB) is always parallel to a radius (AC). Therefore, using Pythagoras’ Theorem:

This is applicable to any planet or spherical object. We have to put in Earth’s data now to get a correct x value. The earth’s radius varies from 6378km at the equator to 6357km at the poles. I’m going to take r as 6370km, an approximate average.

I’m going to take h as 1.6m – my height.

And that’s it. The horizon on a flat plain from ground level is about 4.5km away.

How about from the Britam Tower in Nairobi? The highest floor sits at 135m.

Hardly any difference! You can see about 2m further. Even at 800m (approx the height of the Burj Khalifa) only about 60m further than at ground level.

So how come Kilimanjaro can be seen?

Well, the fact is that neither Britam Tower nor the peak of Kilimanjaro are at sea level. The peak of Kilimanjaro is at a whopping 5895m and Nairobi is at quite a high elevation of 1684m. The diagram is going to have a be modified a bit:

As you can see, I’ve sketched the diagram to show how the heights of our two ‘objects’ protrude from the earth’s surface. I’ve exaggerated their distance apart for clarity.

• r, the radius of the Earth, remains the same as before at 6370×10^3
• k = the height of Kilimanjaro, 5895m
• b = the elevation of the top of Britam Tower, 135 + 1684 = 1819m
• the aim is the find the value of x + y (see diagram), which is the maximum distance the objects could be apart for them to be visible to each other.

We can now break it up into two right-angled triangles, one containing x and one containing y. Applying Pythagoras’ Theorem, we can obtain x and y values as shown:

Now all we’ve got to do is add x and y together:

There’s the answer: 426km. It’s important to realise that this is a maximum value – if the two objects really were this far apart, only the very peak of Kilimanjaro would be able to be seen from Britam Tower. Luckily, at 208km, a lot more of the mountain is visible on a clear day.

If you look back at the diagram, the top line (x + y) would hover above the Earth’s surface, rather than being tangent to it (since the two objects are closer together), in real life – enabling this deeper view of the mountain from Nairobi.