It is important for a sailor to know just how far away land, or another boat might be when first seen. It might be a lot closer than you think!
Because the earth is curved and we can only see in straight lines, known as our "line of sight", we can only see from our eyes to the edge of the horizon or an object sticking up over the horizon.
Where our line of sight hits the horizon is the distance from our boat to the horizon. A small boat first coming into view is assumed to be at that distance. In the picture above, the sailor cannot see the island's palm trees.
If an object, such as a large ship or a palm tree on an island, sticks up well above the water, we can see it from a further distance.
Here is how we can calculate how far we can see from a boat on the sea.
First let's assume that we are sitting in the cockpit of the boat and
our eyes are 5 feet above the water and the seas are smooth. We can
then look at the table below to see that we can see only 2.6 nautical miles
to the horizon.

Now if we are looking at a palm tree on an island, we must determine
how far the palm tree can see, or really how far someone on top of the
palm tree can see. Let's assume the palm tree is 30 feet tall.
Looking at the table, we can see the sight distance is 6.4 nm. To
determine how far that palm tree is from our boat, when we first see it,
we add our sight distance, 2.6 nm, to the "tree's sight distance", 6.4
nm, and find that we can see the palm tree from 9 nautical miles.
This of course assumes no fog, rain, or other conditions that might obscure
our view.
In case you were wondering about other heights, like when the deck of your boat is 4 feet over the water and your eye is another 4 feet over the deck, here is the calculation for finding sight distacne for any height. Visible distance (in nm) = 1.17 x the square root of the height (in
feet)
Try a few other combinations. For example, how far away could
you be and see the first sky scrappers of New York City?
