What is the distance to the horizon at sea?

Question

What is the distance to the horizon at sea?

Answer 1

Imagine the earth is a circle, the center is O

On this circle I have :

my feet are in A

my eyes are in B, the height is h

the horizon line is H

The radius of the earth is R = 6400 km

The angle OHB is right

Thus Pythagore says : OH² + HB² = OB²

or R² + HB² = (R + h)² = R² + 2hR + h²

so the horizon distance is

d = BH = sqrt(2hR + h²)

if I'm on a boat, my eyes are at 2 m over the sea, I see the horizon at :

d = sqrt(2*0.002 * 6400 + 0,002²) = sqrt(25.6) = 5.06 km

if I'm on a mountain (or in a plane) at 1,000 meters, I see the horizon at :

d = sqrt(2*1*6400 + 1²) = 113 km.

But if you consider that the horizon distance is not BH but AH, it's more difficult.

Let t the angle AOH

and I the middle of [AH]

OAH is isoscele, then

sin(t/2) = AI/OA

the formula cos(t) = 1 - 2sin²(t/2) gives :

AI²/OA² = 1/2 (1 - cos(t))

cos(t) = OH/OB = R/(R + h)

then 1 - cos(t) = (R + h - R)/(R + h) = h/(R + h)

AI²/OA² = AI²/R² = h/2(R + h)

AI² = hR² / 2(R + h)

AI = R sqrt(h / 2(R + h))

AH = 2 R sqrt(h / 2(R + h))

AH = R sqrt[2h / (R + h)]

if h = 2 m then

AH = 6400*sqrt(0.004/6400.002) = 5.06 km

(if we consider that the radius is exactly 6400 km !!)

the difference is very small !

if h = 1 km then

AH = 6400*sqrt(2/6401) = 113 km : the difference is very small too.

The difference would be bigest if h is hundreds of kilometers as for satellites

Answer 2

If your eyes are 6' high from ground, about 3.5miles

Answer 3

Less, it's approximately 3 miles.