Computing height vs distance to horizon?

Question

If you know your height above the ground, how far away is the horizon, or how can compute the distance to the horizon?

Answer 1

It's a geometry problem. Suppose you're high up in a building on the beach overlooking the ocean. You're at height h above sea level.

Now draw a circle which represents the earth, with radius r. Your distance from the center of the earth is r+h. The horizon is some distance away, x. Your line of sight to the horizon is to the radius of the earth, forming a right triangle, where the hypoteneuse is r+h.

Using the Pythagorean Theorem,

r^2 + x^2 = (r+h)^2 = r^2 + 2rh + h^2
x^2 = 2rh + h^2
x = sqrt(2rh + h^2)

That's the exact answer, But for practical purposes, 2rh is bigger than h^2. To see this, suppose you're 500 feet in the air. That's about a tenth of a mile, and using miles, h^2 = 0.1^2 = 0.01. On the other hand, the earth's radius is about 4000 miles, so 2rh = 2 x 4000 x 0.1 = 800. This is 80,000 times bigger than h^2.

So we can safely ignore the h^2, and say x = sqrt(2rh). That can be broken down further. Again using miles (you can also do this in kilometers),

x = sqrt(2rh) = sqrt(2r) sqrt h = sqrt(8000) sqrt h = 90 sqrt h

(rounding off, since sqrt 8000 = 89.44; also, h must be in miles).

Using the formula x = 90 sqrt h, suppose you're in an airplane 4 miles up in the air (21,000 feet). Then the horizon is 90 sqrt 4 = 180 miles away.

Okay, now let's work it out for h in feet instead of miles. We'll go back to the formula x = sqrt(2rh), and use r = 3960 miles (a more accurate figure). Then

x = sqrt(2 x 3960 x h / 5280) = sqrt(3h/2) = 1.225 sqrt h

where h is in feet and x is in miles.

Two examples: (1) For h = 6 feet, use x = sqrt(3h/2) = 3 miles. (2) For x = 100 feet, use x = 1.225 sqrt h = 12.25 miles.

To recap: For h in miles, use x = 90 sqrt h. For h in feet, either use x = sqrt(3h/2) or x = 1.225 sqrt h.

In all cases, x will be in miles. And of course, all of this can be done using meters and kilometers.