Enlarge your world: stand on your tip toes

I was at the beach recently, staring out at the cargo ships waiting offshore, and beyond them to the distant horizon, and I thought, how far away is the horizon after all?  Luckily, Phil Plait has the answer to that on his website, it’s a simple question of trigonometry:

Taking the Earth to be 6365km in radius, a man of average height (5′ 9″ or 175.26cm) standing with his toes in the ocean can stare 4.72km out to sea*. The higher up you are, the further your horizon retreats. But while Phil was thinking big – about how far you might see from an airplane or the space station, I’m curious about the subtler effects.  For example, adjusting for the fact that your eyes are in the middle of your head, not the top, knocks a massive 170m off your horizon. In fact, because we’re not very tall, the effect of height is quite powerful:

If you hold your eyes to the ground, your effective horizon is zero metres, and it expands really, really quickly as you stand up. But it’s a diminishing return. In fact, so much so that an extra inch will allow a five-foot-nothing girl to expand her horizons by 38m. Her friend, standing six feet tall, only gets to see an extra 34.5m for the same height boost.

But what if you weren’t just staring out to sea?  What if your world expanded in all directions?  Standing in the flattest place on Earth, Bolivia’s Salar de Uyuni evaporated lake bed, the average man’s visible world covers 70.1 km2.

What happens when you raise your head an inch? You see an extra 34m in all directions. All this adds up!

So my take home point?  I guess it’s that by walking tall, even by just an inch, you get a lot of extra world to see.

* Some caveats: due to atmospheric refraction, weather conditions, ocean swell and tidal surge, the reality will be different. The level of precision described above is not realistic. Pedants are invited to look up the delightful phrase korinthen kacker in their German-English dictionaries.