I was looking at a plane flying by today and thought to myself, how many people right now can see what I am seeing? So my question is, what is the size of the area of earth where one would be able to see a commercial airliner flying at cruising altitude (ignoring daylight issues)? Binoculars/telescopes are allowed.
I think you are missing some key factors in that equation. Population density, terrain, weather, and its size are key among those. Many millions of people could see the space shuttle come in for landing if they wanted when it re-entered the atmosphere and turned back into a plane if they wanted to and had the right equipment and info. That probably isn’t a great example but, if you got all the people in NYC or Tokyo to stand on rooftops on a clear night, the number that could see an airliner on a clear night could potentially be many millions if not hundreds of millions if you brought extra people in for the event.
Since the question is about area, and not number of people, it is a geometry problem. With a sphere with a diameter of 41,851,050 feet, and an object 39,000 feet off of it’s surface, how much of the surface of the sphere could the object “see”? Ignoring atmospheric distortion, and other nasty reality issues, of course. This would give a back of the envelope estimate, anyway.
This is essentially the same question as: from a given altitude, what is the distance to the horizon. The answer is approximately given by the geometric mean of the diameter of the earth and the altitude. If we assume an altitude of about 7 miles and use 8,000 miles for the diameter of the earth, we get about 240 miles. To get the area, use pi*(240)^2, about 180,000 square miles. If there are 100 people per square mile (close to the US average), that would be about 20 million people.
Dunno about planes, but you can use a site like Heavens Above to get a graphical representation of where a satellite is visible at a given time.
Hint: Make sure to set your location under “Configuration”, then click on some of the links under “Satellites”.
Yes it is a geometry problem and thanks for the answers!
Hervé Villechaize never had a problem seeing planes.
If you were at 39,000 feet and flying just south of Long Island, you theoretically could be seen in 13 different states (Maine, New Hampshire, Vermont, Massachusetts, New York, Connecticut, Rhode Island, Pennsylvania, New Jersey, Delaware, Maryland, Virginia, West Virginia) and DC.
That would encompass the following metropolitan areas (with populations from Wikipedia):
New York-Northern New Jersey-Long Island 19,069,796
Philadelphia-Camden-Wilmington 5,968,252
Washington-Arlington-Alexandria 5,476,241
Boston-Cambridge-Quincy 4,588,680
Baltimore-Towson, MD 2,690,886
Assuming there’s no overlap in these figures, that puts you at well over 35 million - and that’s only including the metropolitan areas! There are plenty of people living outside those “metropolitan areas” that would also be able to see you.
For the win!!! 
Bainbridge, NY claims to be within 250 miles of one-third of the nation’s population. So assuming that statistic is correct, flying over that location would theoretically make you visible to somewhere in the ballpark of 100,000,000 people.
Aren’t all of you treating the Eastern Seaboard as if it were some flat part of the Nevada desert? It would be more realistic to take into account the fact that most people’s view of the sky is rarely more than, say, about 30%. (And, where it averages a lot more than that, population densities tend to be lower).
So, I’d cut the above figures down to maybe a third.
Ok, then where would you put the plane to be visible to the most people? I’m just guessing, but somewhere over Southeast Asia? Over Japan? It would be a pretty cool e-map where you could draw a circle anywhere on the globe and get a decent estimate of the population encompassed by it. Maybe the nuclear weapons people have such a thing for casualty estimates based on miles from any arbitrary ground zero.
I think there is a mistake in most of the responses assume if the plane can see the ground, then the people on the ground should be able to see the plane. In practice if you are looking at a plane, even through a telescope, it will only be visible at a considerably shorter distance, since airplanes are much smaller objects than the earth and you are looking at them through a couple of hundred miles of gas, dust and water vapor. It would be equivalent to how far away could a person in a jet see something the size of the house.
I found a thread in an aviation forum and one person claimed that the contrails should be visible from 240 miles with binoculars, but no one claimed the plane itself would be visible.
http://www.airliners.net/aviation-forums/tech_ops/read.main/278538/
And just how many people will be looking up at the sky at any one particular time? I mean, how many people actually go about their day staring up at the skies? There could be flying spaghetti monster battles above the clouds, and how many would actually notice?
At the risk of sounding too much like Dogbert, the OP said “can see a plane” not “do see a plane.”