I swear I have seen this answered already but my search skills apparently suck.
Let’s say there’s a sailboat with a light on its mast 50 feet from the waterline. I’m sitting on Miami Beach at midnight. When the light comes into view, how far away is the boat?
If you want an added bonus, give me the final answer and put the trig in little tiny letters after 
As as rule of thumb, the distance to the horizon (in miles) is about the square root of the height (in feet). If your eye is 5 feet above the water, thats about 2.2 miles to the horizon. For a light at 50 feet, that’s about 7 miles beyond your horizon. Adding the two, you get a bit over 9 miles to the sailboat.
To get more precise you’d need the expact formula plus an adjustment factor for atmospheric pressure & temperature.
About 9 miles. But that neglects atmospheric effects which cause light to bend around the earth to some extent. So the boat could be farther out.
Here’s the trig. Draw a circle (the Earth), and draw a radius. Now draw a tangent to the circle at the point where the radius intersects it (A, where you are sitting on the beach). Pick a point a bit out from you on the beach (X) and draw a line from the center of the circle that intersects your tangent. The tangent is your line of sight and the place it intersects the line you just drew (B) is the top of the sailboat’s mast. The center of the circle is O.
The triangle OAB is a right triangle, the right angle being at A. The sides are OA = R (the radius of the earth), AB = d (the distance to the sailboat), and OB = R + h (h is the height of the mast).
Pythagoras:
R[sup]2[/sup] + d[sup]2[/sup] = (R + h)[sup]2[/sup]
Expanding, = R[sup]2[/sup] + 2Rh + h[sup]2[/sup]
Cancel the R[sup]2[/sup] terms
d[sup]2[/sup] = 2Rh + h[sup]2[/sup]
Now notice that the h[sup]2[/sup] term is a LOT smaller than 2Rh, because the sailboat mast is miniscule compared to the radius of the Earth. Throw away that term.
d ~= sqrt (2Rh)
R = 3963 miles * 5280 feet / mile
h = 50 feet.
d = 45743 feet = 8.66 miles
Distance to horizon is square root of twice the height of your eye level times the radius of the globe.