When you look up in the sky and you see a contrail, how far away is it? How far away are these contrails, a mile? two miles? Would you believe they are actually 20 to 100 miles away?

Contrails typically form above 30,000 feet, or around six to eight miles straight up. It’s quite hard to judge exactly how high it is unless you know what type of plane it is. But assuming six miles is a pretty safe bet.

So if we assume it’s at least six miles above the ground, then all we need to do to find out how far away it is is to measure the angle of elevation. It’s quite a common child’s math problem but usually applied the other way around, to measure the height of things if we know how far away they are.

So it’s a really simple bit of maths, height = distance * tan(angle).

But of course, if you know the height, then you can calculate the distance = height * (1 / tan(angle)).

Now most planes leaving contrails cruise at between 30,000 and 45,000. 6 miles is 31,680 feet, so it’s a pretty safe assumption that any contrail you see is at 6 miles or above. So for the purposes of calculating the distance, let’s just assume it’s six miles high. At worst we will underestimate. Here’s what the figures work out as for various angles from the horizon:

Angle 1/TAN(Angle) Height Miles LOS Sq Miles km Sq Km 5 11.4 6 68.6 68.8 14776 110.4 38269 10 5.7 6 34.0 34.6 3638 54.8 9421 15 3.7 6 22.4 23.2 1575 36.0 4080 20 2.7 6 16.5 17.5 854 26.5 2211 25 2.1 6 12.9 14.2 520 20.7 1347 30 1.7 6 10.4 12.0 339 16.7 879 35 1.4 6 8.6 10.5 231 13.8 597 40 1.2 6 7.2 9.3 161 11.5 416 45 1.0 6 6.0 8.5 113 9.7 293 50 0.8 6 5.0 7.8 80 8.1 206 55 0.7 6 4.2 7.3 55 6.8 144 60 0.6 6 3.5 6.9 38 5.6 98 65 0.5 6 2.8 6.6 25 4.5 64 70 0.4 6 2.2 6.4 15 3.5 39 75 0.3 6 1.6 6.2 8 2.6 21 80 0.2 6 1.1 6.1 4 1.7 9 85 0.1 6 0.5 6.0 1 0.8 2 90 0.0 6 0.0 6.0 0 0 0

The important column there is in bold, the “Miles” column which tells you how far away horizontally the plane is. That is, it tells you how far away the point is that is on the ground directly below the plane (ignoring curvature of the earth, and local changes in height).

The column next to that, LOS (Line Of Sight) tells you the actual distance from you directly to the plane. So when it’s overhead it’s 6 miles (straight up). As it gets further away it gets closer to the horizontal distance.

Look at the angles below 45 degrees. In particular, a plane that’s ten degrees above the horizon will be 34 miles (55 km) away, and one that is five degrees will be 68 miles (110 km) away. Now ten degrees might not seem like a lot but it actually is surprisingly high if you go out and point your arm up at ten degrees.

Here I’m pointing up at ten degrees, I see plenty of planes “over there”, it seems pretty unintuitive to think that they are 50 miles away. But they are. Notice (as you would expect) at 45 degrees, the plane is the same distance away as the height, six miles. And any angle above 45 degrees is within a six-mile radius. But still, consider that many people will point to a plane at 45 degrees as being “overhead” when really it’s six miles away.

Now let’s look at some actual contrails. These photos were taken on an iPhone using the “Theodolite” app, which tells you the angle of elevation.

This math breaks down somewhat for below five degrees, as they the curvature of the earth is much more of a factor. But for most contrails, you see you can get a reasonable estimate of the distance from the angle. And since most contrails do not fly directly overhead, the vast majority of visible contrails are going be at a low angle, and very far away. Much further than you might think.

Back to the photo we started with. It was taken from a bridge over the Thames to the west of London. Using Google Earth, I’ve placed a grid at six miles altitude, with five miles between each line.