# measurement of horizon?

A person of height 1.35 meters is looking toward the horizon. The radius of the Earth is 6380000 meters. What is the distance from the person to the horizon in meters?

Relevance

If this is a homework question in geometry, then draw the triangle.

The height of the person's eye = radius of Earth + height of the person.

Call it Rh = R + h = 6 371 000 + 1.35

Rh = 6371001.35

The horizon is the point where the person's view is tangent to the surface; the line from the person's eye to the Earth's circumference makes a right angle (90 degrees or pi/2 radians, as you wish) with the line from Earth's centre to that point (therefore, the radius.

This gives you a right-angle triangle with the hypotenuse being Rh and the side from Centre to surface being R.

Cosine angle = adjacent / hypotenuse = R / RH

This angle is at Earth's centre.

Cos(angle) = 6371000 / 6371001.35 = 0.999999788...

The arcCos of that (inverse of cosine) = an angle of 0.000650996 radians

Because a radian is a fraction of the radius, measured along the circumference.

The distance to the horizon =

6371000 * 0.000650996 = 4147.4966 m

or

4.147 km

Before the invention of electronic calculators, we did not bother with such a calculation. We relied on approximations. The one I remember is

Distance to horizon (in nautical miles) = 1.1 sqrt(height in feet)

A nautical mile is 1862 m

More recent approximations (as found in other answers) do take into account the air refraction (the horizon you "see" is a tiny bit further out, because the air "bends" the path of light from that point to your eye). That is why 4.48 is not a bad answer.

• Around 15, 000 Meters or 15 Kilometers to be correct

11 Miles Source(s): And proof if needed for certain Idiots that the Earth is a Globe
• Distance to horizon is d = 1.323√h

d in miles and h in feet

or d = 3.856√h

d in km and h in meters

d = 3.856√1.35 = 4.48 km

• Anonymous
1 month ago

Bring me the horizon and I shall measure.