Puzzle Time – Distance To Horizon

A couple of months ago I was on a flight and had a thought: How far is the horizon at this very moment?

I got my notebook and pen out and tried to solve this using only the knowledge I had, since there was no internet anyway. Of course, many assumptions were made as were convenient for the calculation.

I planned on checking my answer once I landed. If you want to attempt it, stop reading here and come back later when you have a solution. Unfortunately for me, I made a little error subtracting between two big numbers and got a completely wrong answer.

Assumptions

I assumed the following

  • Perfect visibility, no degradation due to atmosphere
  • Earth is a perfect sphere with a radius of 6400km
  • Aeroplane flying at an altitude of 10km

Solution

Thanks to the assumptions, the problem reduces to primary school geometry problem and is trivial to solve once visualized clearly on a piece of paper.

Using Pythagoras Theorem we know how to calculate the third side of a right angled triangle give the first two. We also know that a tangent to a circle (the line of sight from an aeroplane) meets the circle in a way that it is perpendicular to the radius drawn from the point of contact.

Combining the two pieces of knowledge, the problem can be trivially arranged as follows

(R + r)² = R² + Dh²

Where

  • R is the radius of the earth (in Kilometers)
  • r is the distance of the aeroplane from the surface of the earth (in Kilometers)
  • Dh is the distance to the horizon

Rearranging, we get

Dh² = (R + r)² – R²

Expanding the (a + b)² equation

Dh² = R² + 2Rr + r² – R²

Dh² = 2Rr + r²

Taking square root on both sides

Dh = √(2Rr + r²)

Substituting the values

Dh = √(2*6400*10 + 10²)

Dh = √(128000 + 100)

Dh = √(128100)

Dh = 357.9

Distance to horizon is about 358 kilometers when the aeroplane is at an altitude of 10 kilometers. Now looking at the map while sitting in a flying aeroplane is a lot more insightful as I can guess which cities I’m looking at outside the window.

That’s it for this little post. Thank you for reading!

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