What is the mathematical formula for finding the length of a line
segment that goes from the top of a line segment that is perpendicular
to the surface of a circle to the tangent of the circle? In other
words, if a 6 foot tall person was in the desert, how far away would
the horizon be? I would like the answer and the formula.

RDowney111 --

The simple answer is about 6.2 km or 3.4 miles, if you're 6' tall and
4' above the waterline on the beach or 10' above the desert surface. 
If you're 6' tall and standing with your feet at the edge of the shore
or desert, it's about 5 km or 3.1 miles.


The derived formula
is indicated on this web page:
Goddard Space Flight Center
"Distance to the Horizon" (Stern, Dec. 13, 2001)
http://www-istp.gsfc.nasa.gov/stargaze/Shorizon.htm

The formula is:
D = 112.88 km * h^1/2

Or 

D = 112.88 km * SQRT h

Where:
D = distance to the horizon in kilometers
112.88 is actually the square root of the earth's diameter in km
h = height in kilometers

Since 10' is about 3 meters or .003 km, D = 6.18 km.

---

Here's an alternate formula and way to calculate it on the "How Stuff
Works" page.  The diagram is excellent, but since it mixes feet and 
miles; meters and kilometers, it's a little more confusing:
How Stuff Works
" When I stand at the water's edge and look out over the ocean, how
far away is the horizon?"
http://www.howstuffworks.com/question198.htm

---


Now some caveats:
*  Note that you see a ship that's further offshore because it's
taller, rising above the horizon -- a fact known to seafarers even in
Columbus' time (debunking the stories that Columbus' men believed in a
"flat earth").
*  atmospheric conditions can refract light over the horizon. 
Temperature inversions will have that effect.  It's also known in a
phenomenon called "looming".  Author Nathan Philbrick references it in
his book "Sea of Glory," written about the Wilkes Expedition of
1838-1842.   I've seen it on Lake Michigan and it's apparently quite
common in Antarctica, as Philbrick says maps created by the Wilkes
Expedition made some errors in judging
distance when it made the first maps of the Antarctic coast.  Here's
an explanation of looming:
SDSU
"Distance to the Horizon" (Young, 2003)
http://mintaka.sdsu.edu/GF/explain/atmos_refr/horizon.html

Best regards,

Omnivorous-GA

Excellent research, complete answer with references, and timely given. Thanks!

Here's another way using just the Pythagorean theorem: imagine a right
triangle, where the hypotenuse ("c") is the distance from the center
of the earth to the top of the six-feet tall person's head at sea
level and one leg ("a") is the distance from the center of the earth
to the interception with the tangent line (that's where you see the
horizon).  Since a^2 + b^2 = c^2, and we know that the earth's radius
is 6,378 km, we plug in 6,378 for "a," 6,378.0018288 for "c" (since we
have to add the height of our six-feet tall person, and six feet
equals .0018288 km), and solve for "b."  We get 4.83 km (the distance
from the top of the guy's head to the horizon), and that's about 3
miles, as omnivorous said.

The last comment does not seem right. The pythagorean theorem requires
a right triangle. Which angle is 90 degrees in your example ipfan-ga?

Actually, ipfan-ga is correct (in fact, he is just restating the
original answer). See the diagram and the second paragraph at:
http://www-istp.gsfc.nasa.gov/stargaze/Shorizon.htm