## where are my horizons?

Recently, while standing at the beach on a calm day, I was asked how far away the horizon is. How far can we see? How far away is a ship on the horizon and is the earth curved enough that we would see the top of it first as it came up around this ball we call Earth?

I didn’t know the answer. But I was pretty sure math did.

To get a quick idea of the answer we need one number with a simplifying assumption: the earth is a sphere with a radius of 6378.1 km.

I will show my work for those who are interested. Those who are not interested can scroll down to the bottom.

Nerds love diagrams. Here is mine.

$r$ is the radius of the earth
$h$ is the height of the person looking out over the smooth ocean
$L$ is arc of the earth’s surface over which the person looks
$\theta$ is the angle subtended by the arc

geometry gives us a relationship between the sides of our triangle

$\cos{\theta} = \frac{r}{r+h}$

the length of an arc is proportional to the radius of its curve and the angle it subtends

$L = r\theta$

substitute!

${L} = {r}\cos ^{ - 1} \frac{r}{{r + h }}$