From a height of six feet above sea level, looking out onto a clear sea, how far away is the horizon?

According to this web-page the calculation is made by finding the square root of the height of your eyes (in feet) and multiplying that value by 1.17.

The square root of 6 is 2.45; multiply that by 1.17 and you get an answer of 2.86 nautical miles.

https://www.boatsafe.com/calculate-distance-horizon/

Or 3.29 real miles!

The situation involves a right angled triangle involving the distances from the eyes to the centre of the Earth (R+h), from the horizon to the centre of the earth (R) and the required distance to the horizon x. Suppose all distances are in feet.
By Pythagoras, x = Sq rt ((R+h)^2 - R^2)
= Sq rt (2Rh + h^2)
Given that R is the radius of the Earth, then h^2 is so small compared to 2Rh ignoring it won't affect our approximate result.
So x ~ Sq rt (2Rh) = Sq rt (2R) * Sq rt h
The radius of the Earth is about 3,960 miles or 3,960 * 5,280 = 20,908,800 feet.
So Sq rt (2R) ~ 6,467.
So x in miles ~ 6,467/5,280 * Sq rt h ~ 1.22 * Sq rt h.
For h = 6 ft, I make x to be as near as dammit 3 miles.

I guess what I'm trying to say is yay, Mutch!

3,571.59 metres

This site is useful for calculating the distance at any given height above sea level.

Ref: http://newton.ex.ac.uk/research/qsystems/people/sque/physics/horizon/

Distance to horizon
In order to determine how far away the horizon is for any given observer, first determine how far above sea level the observer's eyes are. If the observer is standing on the surface of the ocean, his eyes might typically be somewhere between 140 and 200 centimeters above sea level. Look this number up in the first coloumn in the below table. The second coloumn then tells you how far away the horizon is.

Note that obstructions or poor visibility might further restrict how far an observer can see. The distance to the horizon is simply an upper limit to how far away it is physically possible to see - other circumstances may very well restrict this much further.

Example Ut'kikk has been placed in the tops of a sailing ship to keep a look out. The skies are clear and visibility is excellent. Ut'kikk is currently located up in the main mast, 12 metres above sea level. He can see as far away as 13 kilometers (approximately).

pvv.ntnu.no/~bcd/shadowworld/info/horizon.html no longer exists

This is dependent upon where on the Earth one is. The Earth IS an irregular Sphere and thusly Horizon will vary from Equator to Pole.

Actually, the variation in curvature of the Earth at various points is too small to be greatly significant.
Note that the calculated distance to the horizon will be more than the actual distance due to the refraction of light through the atmosphere.
https://en.wikipedia.org/wiki/Horizon

AT SEA:

As you can see by the following, it varies (in part) by the height of the eye:

Height of eye (specify units): 5.6 (Decimal)
feet meters

Distance to the Horizon: 2.7663589065773806 (Nautical Miles) 3.1852173552208334 (Statute Miles)

Height of eyes: 5.0 feet
2.6139634656972545 Naut. Miles
3.0097474977147174 Stat. Miles

Calculator for sea:

http://www.boatsafe.com/tools/horizon.htm

Presuming you are standing at sea level, not on an elevated piece of land, and you have an eye-level of 1.7 metres, then, ignoring possible hindrances to line-of-sight, the horizon will be 4.7 kms distant.

https://en.wikipedia.org/wiki/Horizon#Distance_to_the_horizon

The horizon is where you see it, and a six-foot-tall man has eyes at about 5-1/2 feet. When you factor in the curvature of the Earth (remember it's spherical, not flat!) this is your answer:

SqrRt(man's eye-height in ft / 0.5736) = horizon's distance in mi
SqrRt(5.5 / 0.5736) ~= 3.1 miles

The formula will vary depending on the units of measure you use.

http://science.howstuffworks.com/question198.htm