Associated Topics || Dr. Math Home || Search Dr. Math

### 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.

Tony
```

```
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
problem.

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
far):

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
this?

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

Search the Dr. Math Library:

 Find items containing (put spaces between keywords):   Click only once for faster results: [ Choose "whole words" when searching for a word like age.] all keywords, in any order at least one, that exact phrase parts of words whole words

Submit your own question to Dr. Math
Math Forum Home || Math Library || Quick Reference || Math Forum Search

Ask Dr. MathTM
© 1994- The Math Forum at NCTM. All rights reserved.
http://mathforum.org/dr.math/