The distance to the horizon is simple to calculate if you know the radius of the sphere your standing on and the height of your observation. Several Web references have Curvature often quoted as “8 inches/ mile” . This is an accurate number but is often misunderstood as to its meaning.
For arguments sake we will assume the earth is a smooth sphere of 3950 miles radius. We will wipe out waves, tides, hills and valleys to simplify the calculations and make the concept simpler to visualize and therefore easier to understand. We will also remove atmospheric fog and temperature diffraction so there is no misunderstandings.
With an 8 inch / mile curve you can start anywhere , walk a mile in any direction and you will be 8″ lower than from where you began. I say lower but in fact you will be exactly the same “height” or distance from the center of the earth. No matter how far you walk you will always be on the surface and therefore always the same height.
Lets have a look at the visual limit. This is how far you can see before the Surface of the earth drops out of view due to its curvature. This “limit” can actually be described or measured in 3 ways, the horizontal distance, the line of sight (LOS)distance, and the walking distance. For small heights these are all virtually the same. The horizon calculation requires some simple Pythagorean math and a the little picture below to clarify the meanings of the distances involved.
Below is a chart of the “Drop” at different distance out or over. I have also shown the gravitational plumb or “Tilt” of the object as one moves away from the starting reference point. We will notice that the quoted “8 inches/ mile” curvature is not really useful here in establishing how far you can see. It applies roughly only to the first 8″.
Looking at the graphic, you can see why at “3950 miles distance” the drop is 3950. Its a radius. This chart has nothing to do with how far you can “see” but are numbers basically defining the curve of the earth. You can also see that in order to calculate the horizon distance, the viewers height above the ground and planet radius are the only numbers needed.
If your eye is 5 feet (60″) off the ground the horizon (tangent edge of the sphere) will appear to be 2.74 miles away. That’s as far as you can see. You can see beyond the horizon only if the object you are looking at has some additional height of its own.
If we have a ladder to climb up we get a better vantage point and can see a bit farther. If we are on a ship adrift in a large ocean, the crows nest offers a much farther view than does standing on the deck of the ship. I suppose that’s why they built them in the first place.
If we stand on the ship deck our eye is say 32 feet off the surface of the water. From here, we can see the world out to about 7 miles in any direction. Lets climb the crows nest which is 100′ higher than the deck. Now at 132′ view point ,we can see out 14 miles, or an additional 7 miles to the horizon. This is 2 times the horizon distance and 4x the area of ocean now within view.
Lets imagine we are looking for a volcanic Island that rises 500 feet out of the ocean. The top of the island can “see back to its horizon” about 27 miles. Adding this to our crows nest horizon limit of 14 miles, we find we should be able to spot the top of the island from 41 miles away. “Land Ho!”
We could take an airplane, a balloon ride (30,000 ft = LOS of 212 miles) or the view expected from an orbiting satellite. All are easy to calculate the horizon distance. The free desktop version of “Google Earth” has a readout of your observation height and the view of the planet from that point (out to 63,000 km) and even the curve of the horizon can be see. Technology is wonderful!
Below is a quick reference chart of ” How high, how far..” showing some different vertical vantage points compared to how far the horizon appears. The right column is your view limit to the horizon for something that is flat. If the object your viewing has height, you must add its “reverse” viewing distance to your own.
You will notice that even as the vertical viewing distance increases drastically, you can never see half of the earth (3950.0) . Your missing an inch or two on the edges.
( some abbreviations in the above chart… leo = low earth orbit, GS = Geo stationary orbit. moon and sun distances are approximate.)