How to Calculate the Distance to the Horizon
Have you ever been out on a leisurely cruise and suddenly wondered, "How far it is to the horizon?" Or maybe your destination is a port that has a lighthouse and you wonder "How far away will I be when I see the lighthouse?" (Well, you're in luck, even if you are a sick unit that thinks of these sorts of things - so are we.) We have the answer!
Of course you can find tables that do the calculation for you in numerous navigation books, almost every book that talks about passagemaking, the Coast Pilot, almanacs, etc. But what if you didn't have any of these references onboard? How could you calculate the distance to the horizon or the "distance off" if you know the height of an object?
It's simple, really. If you want to know the distance to the horizon you simply have to know your height of eye. That is the distance that your eyes are off the surface of the water.
If you're in a jon boat, that would probably be about three feet (if you are sitting like you should be in a jon boat). Of course, if you were in a jon boat you probably wouldn't care how far the horizon was.
Anyway, I digress. If you are on the tuna tower of a sport fishing boat you may be 15, 20, 25 feet above the surface of the water.
Once you know your height of eye you simply plug that into the following formula:
1.17 times the square root of your height of eye = Distance to the horizon in nautical miles
For example, if your height of eye is 9 feet above the surface of the water, the formula would be:
1.17 times the square root of 9 = Distance to the horizon in nautical miles.
1.17 * 3 = 3.51 nautical miles
If you want to calculate the distance at which an object becomes visible, you must know your height of eye and the height of the object. You then do the same calculation for your distance to the horizon and the object's distance to the horizon and add the distances together. For example:
You have the same height of eye of 9 feet so your distance to the horizon is still 3.51 nautical miles. You're approaching a port that has a lighthouse that is shown on your chart to have a height of 81 feet. Using the same formula you would find that 1.17 times the square root of 81 (1.17 * 9) = 10.53 nautical miles (the light house can be seen 10.53 nautical miles over the horizon)
By adding the two together: 3.51 + 10.53 = 14.04 nautical miles, you should be able to see the lighthouse when you are 14.04 nautical miles away.