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How Far Can You See?

Assuming an Earth-sized planet (and why not?) the distance to the horizon is according to the following formula:

D = √(1.5*H)

D is the visibility distance in unobstructed miles
H is the height (in feet) being viewed from

So, a 6' tall man can see about 3 miles in every direction if the terrain is flat and featureless, such as the ocean or the famed Plains of Billiard. A man on a warhorse, with his eyes at 9' above the ground, could see about 3.7 miles. A sailor 60' up in a crows-nest could see about nine and a half miles.

If you want to know how far away the same 6' tall man could see a 300' tower, just work out how far the horizon is at 300' (about 21 miles) and add the bloke's 3 miles -- that's 24 miles before the tip of the tower falls below the horizon. (That doesn't take into account atmospheric haze, astigmatism, etc.)

For non-Earth-sized planets, the formula is

D = √(2*h*R)

where R is the radius of the planet.

(You need to be using a rational measurement standard here, like metres).

So, you're on a tiny planetoid just 1,200km (1,200,000 metres) in diameter. You're 1.5 metres tall. You can see the square root of (2*1.5*600,000) metres, or 1.34 kilometres. You can just see the top of the head of somebody the same height as you about 2.6 kilometres away.