What is the distance to the horizon?

Answer 1

The distance to the horizon depends on the altitude of the observer. Assuming that the earth is a sphere and that the horizon is at sea level, the geometry of the situation involves a right triangle with the hypotenuse equal to the radius of the earth plus your altitude (R + A) and the opposite side equal to the radius of the earth (R). Then it's only necessary to apply the Pythagorean theorem to calculate the third side:

D = sqrt( (R + A)2 - R2 ) = sqrt( 2RA + A2 )

where the radius of the earth is 6367.5 km or 6.3675 x 106 m. If you're standing on the beach looking out to sea, then A is at most a few meters, say 3 m as an example. In that case, the horizon is about 6181 m or 6.181 km away. If you're on a high bluff or on the roof of a tall building looking out to sea, and your altitude is 150 m, then the horizon would be 43,706 m or 43.706 km away. The arithmetic is slightly more complicated if the horizon is anything other than the ocean.

Another Way To Look At It:

A simple approximation formula:

The distance in miles, is the square root of one and a half times the height in feet.

So, for a six foot tall person, standing on a beach: 1.5 times their height is 9. The square root of 9 = 3. So, the horizon is about 3 miles away.

Note: This simplified formula is quite accurate, but only works if the height is in feet and the distance in miles. If you want to use height in meters and distance in kilometers, you need to change the multiplier from 1.5 to 12.74.