Russell Chalmers of Dunedin asks :-
How far is it from the shore line to the horizon?
John Campbell, a physicist with an interest in atmospheric optics, responded.
About 4.4 km, depending on how tall you are.
This can be easily worked out by anyone who can work with the geometry of a right-angled triangle (Pythagoras' theorem). First let us assume the Earth is a perfect sphere of radius R, our eye is a distance H above the surface (shore), and the straight line distance from our eye to the horizon is L. Then, using the triangle which has a right angle at the contact of the horizon, (R + H) multiplied by (R + H) = (L multiplied by L) + (R multiplied by R). If the height of our eye above the shore is very small compared to the radius of the Earth (about 6378 km), then we can ignore small terms and L is given by the square root of (2RH). Hence, for the Earth, a rule of thumb is the distance in km to the horizon is the square root of (13H), where the height of the eyes above the shore line is given in metres. If you are standing on the shore and your eyes are 1.5m above the shore then the horizon is just over 4.4 km away.
Sorry about the mathematics but it is interesting to have the rule of thumb. Tall people see a more distant horizon than short people. In sailing ship days, when human vision was the only way to locate a distant ship or other object, whalers and others always had look-outs as high in the masts as was feasible, hence the name - the crows nest.
Keep in mind that if we see the masts or superstructure of a ship on the horizon but we do not see the hull, then the ship is further away than the sea horizon. If we are on land the geometric horizon may be hidden by trees, buildings or mountains. For example, when standing in a large flat crater the rim will be a lot closer than the horizon would be if we were standing on a very large, very flat, plain.
Also we have determined the straight-line distance between our eye and the horizon. If we wish to know the distance around the curved surface of the Earth that is a much harder problem mathematically, but for small heights above the shoreline it is much the same as the direct distance. For example it is only 1cm shorter (in about 11.3 km) if the eyes are 10 metres above the shore level.
Note that we have only considered a perfect Earth. In practice, it isn't quite spherical but has a bulge at the equator. Also there is a small tidal bulge of a metre or so. Large waves can be several metres high so they too affect the level of the horizon. If the air above the sea is warmer or colder than the sea then light travels not in a straight line but is deviated, as it is in a mirage or a looming image. Hence the distance to the horizon depends slightly on the day, the place, the time and the weather.