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Distance to the Horizon

Date: 09/18/97 at 23:08:17
From: Tony Turner
Subject: Geometry/Trig

If a 6-foot man is standing on the beach at sea level looking straight 
out to sea, how far can he see - i.e. what is the distance from the 
man to the horizon?  

The radius of the Earth is approximately 3963.21 miles. Also, this 
question assumes the Earth is perfect circle.

We have been arguing over this question for two months at work. Using 
a squared + b squared = c squared, we came up with 3 miles. We know 
this is wrong because you can see farther than 3 miles.  

Any help would be greatly appreciated.


Date: 09/22/97 at 16:35:32
From: Doctor Ken
Subject: Re: Geometry/Trig

Hi Tony.

You're right, you need to use the Pythagorean Theorem to solve this 

If you let the radius of the Earth be R, and the distance your 6-foot 
man can see be D, then you get this formula (I assume you got this 

  R^2 + D^2  =  (R + 6)^2

Which simplifies to:

  D^2  =  12R + 36
  D^2  =  12*20925748.8 + 36     (plug in radius of earth in feet)
  D^2  =  251109021.6            (do the arithmetic)
  D = Sqrt(251109021.6) 
  D = 15846.4 feet
  D = about 3 miles

So you were right. But note that this actually tells you how far away 
the farthest point _on_the_surface_ of the Earth is. If you're trying 
to see some other 6-foot person standing on a raft way out in the 
ocean, he/she can actually be 6 miles away (you'll only be able to see 
the top of his/her head, though).  Can you draw a picture to justify 

An interesting exercise for you might be to come up with a formula 
that tells you how far away you can see another object - given your 
height and the height of another object, how far away can the other 
object be while you can still see it?  Again, your solution would use 
the Pythagorean Theorem.

Another interesting thing to try would be to come up with a good 
approximation to your formula that's easier to compute in your head.  
If you can come up with something like this, it will come in handy in 
mountainous regions - if two mountains are 9 miles apart and you can 
see the top of one from the top of the other, what can you say about 
their height?

-Doctor Ken,  The Math Forum
 Check out our web site!  http://mathforum.org/dr.math/   
Associated Topics:
High School Geometry
High School Triangles and Other Polygons

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