Hello, TWoods!

How far can you see to the horizon?

The circle represents a cross-section of the Earth.

Imagine you are in a balloon at $\displaystyle B.$

$\displaystyle BT$ shows the furthest point, $\displaystyle T$, on the Earth's surface that you could see. Code:

B
o
|\
| \
h| \
| \
| \
* * * \
* | *\ T
* | o
* R| * *
| *R
* | * *
* o *
* O *
* *
* *
* *
* * *

The balloon is at $\displaystyle B,\;h$ miles above the Earth.

The center of the Earth is $\displaystyle O$; the radius is: $\displaystyle OT = R$

The angle at $\displaystyle T$ is 90°.

Using Pythagorus on $\displaystyle \Delta BTO\!:\;\;BT^2 + OT^2 \:=\:BO^2 \quad\Rightarrow\quad BT^2 \:=\:BO^2 - OT^2$

. . $\displaystyle BT^2 \;=\;(R+h)^2 - R^2 \;=\;2Rh + h^2$

Therefore: .$\displaystyle d = BT$, the distance from the balloon to the horizon

. . is given by: .$\displaystyle \boxed{d \;=\;\sqrt{2Rh + h^2}}$ miles

where: .$\displaystyle h$ = height of balloon, $\displaystyle R$ = radius of the Earth (both in miles).