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.

viewable diagram

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 }}

For math this gnarly you will need to set your calculator to rad (aka “radians”).

I am 183 cm tall, so my horizon is about 4832 m away. Less than 5 km, wow.

If you don’t want to pull out your calculator, you can just look your height up here.

Viewable vs height

And if you build a lookout post or climb to the crow’s nest on the mast of your very tall ship…

Viewable vs altitude

Now that the math is done, you are equipped to answer those questions at the start. Let me know what you decide.

Double your fun: you could probably also use this approach to talk about how far away a tractor is in Saskatchewan.

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