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Boating Center - How far is the horizon ?
When I was a kid, whenever we went "down the shore" to Avlon, NJ, I would stand at the water's edge hoping I could see England. (Geography was not my favorite subject!) I never could see England from NJ, but often would see large ships on the horizon and wonder how far away they were. I didn't know at the time that there is a formula for figuring out just that.
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 sail boat, that would probably be about three feet (if you are sitting like you should be in a sail boat). If you were standing at the water's edge, your height of eye is the distance from your eye to your feet.
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, let's say you are on the water in a friend's sport fishing boat and your height of eye is 9 feet above the surface of the water. The formula to calculate distance to the horizon is:
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.
As you can see from the illustration, because of the curvature of the earth, the higher your height of eye the farther you can see, and vice versa.