From Fisherrow Harbour, you can look across the Forth Estuary and see the waterfront at Kirkcaldy (10 miles away). But if you look further up the coastline, it’s impossible to see the beach at Leven (18 miles away) even on the clearest day. We live on a curved earth, but is the effect really “strong enough” to allow you to see something 10 miles away but not 18 miles?

Here is the world, with our two metre tall observer looking out towards the horizon. How far away is the horizon?

Firstly, let’s figure out what we know already. The observer’s eyes are 2m above sea-level. The observer’s feet are 6371km from the centre of the earth, and the horizon is also the same distance from the centre of the earth. Finally, the direction that observer is looking makes a right angle to the line joining the horizon point to earth’s centre. This means we have a right angle triangle:

The long side of our triangle has length r (earth’s radius) plus t (how tall the observer is). The red side has length r (earth’s radius). We don’t yet know how long the other sides (‘d’ for distance) is yet. But Pythagoras Theorem tell us how to calculate it (see image).

Let’s do some concrete examples:

1) If we are 2m tall, the horizon will be about 5km away. This also means that we’d be able to just see the eyes of a 2m tall person who is 10km away.

2) If we are 5m tall (eg. 2m tall person standing on a 3m high pier), the horizon is 8km away. We’d be able to see the top of a 5m tall structure that’s 16km away.

Let’s return to our original question. Kirkcaldy’s waterfront is 10 miles or 16km away. Kirkcaldy is visible from Fisherrow because both you and the structures you’re looking at are 5m above the water. But the beach at Leven is 18 miles or 29km away. It is hidden by the curvature of the earth. If you wanted to see something at that distance, both you and the structure you’re looking at would need to be 18m tall. The beach at Leven certainly isn’t that high. But there is one nearby object which is tall enough and consequently visible – the Levenmouth Test Turbine at a whopping 110m tall.

(In practise, the atmosphere also bends light which slightly increases the distance you can see)

On a boat, there’s two situations where this knowledge is relevant – observing what’s around you, and using your VHF.

Visual observations Let’s assume that when you stand on a boat, your eyes are 2m above sea level. We’ve already seen that the horizon will be 5km away, and you’d be able to see the eyes (but not the boat) of another person 10km away. So if you are keeping watch for small vessels, then 10km is about the maximum distance you could hope to see.

So far, we’ve only dealt with the case where the observer and the structure being viewed are at the same height. We can use the same maths to find out how far away a structure of arbitrary height be seen.

The Maran Artemis is an oil tanker sailing up the Firth of Forth just now. It is 336m long, and roughly 35m vertically of which 10m is under water. So it’s about 25m tall above the water. If we’re sitting on our little boat at 2m above sea level, we already know the horizon is 5km away. Let’s now imagine an observer sitting at the top of the Maran Artemis. Being 25m tall, for them the horizon is nearly 36km away. If we place our boat on one ‘side’ of the horizon, and the Maran Artemis on the other ‘side’ then we have a total of 41km (25 miles) – being the distance that we’d be able to just see the top parts of an oil tanker.

VHF radio VHF is ‘very high’ frequency radio waves, and they travel in mostly straight lines (they can’t “bend” around obstacles as readily as low frequency waves can). So, we can use the same reasoning we used above to calculate the distance that our VHF will work. This won’t be exactly correct, since we’re not taking into account power, antenna efficiency or atmospheric effects, but it’s a useful guide.

If our mast-top VHF aerial is 10m above the water, it’ll be able to “see” a similarly mounted VHF aerial 22km (14 miles) away. Beyond that the curvature of the earth gets in the way. So that’s the limit if you are trying to talk to another boat. (In reality, you can get a little further because VHF does bend slightly, but your signal strength drops quickly as distance increases).

The coastguard are clever people. They want to be able to talk to boats as well as possible. Consequently, they make sure their aerials are mounted as high as possible. For the Firth of Forth, the closest coastguard aerial is on the Craigkelly tower just behind Burntisland. The tower itself is 125m tall, and it stands on a 182m hill, making it 307m in total. If you were to climb to the top of the tower, the horizon would be 62km away. And if our little boat with a 10m mast was 11km beyond that horizon we’d just be able to see it. That gives the coastguard the capability to ‘see’ boats that are 73km (45 miles) away from the Craigkelly transmitter.

To conclude The curvature of the earth is easily visible in the Firth of Forth – you can see Kirkcaldy at 10 miles, but not Leven beach at 18 miles. If you are on a small boat, you can only see other small boats up to 10km (6 miles) away and tankers (the very top part) up to about 30km (20 miles) away. Your VHF can only ‘see’ other boats up to about 22km (14 miles) but will easily see the Craigkelly transmitter up to 73km (45 miles) away.