I saw an animation this morning showing how the space above our planet is dangerously crowded with satellites. That motivated me to do a little back-of-the-envelope math.
The vast majority of satellites are in low earth orbit (LEO), which extends from 160 to 2000 km above the earth’s surface. The radius of the earth is about 6400 km, so the volume of the LEO region is
There are about 12,500 satellites in LEO, so the average volume of LEO per satellite is about 100,000,000 km³.
Now this isn’t the last word in collision avoidance—there are lots of complications we’re not going to get into here—but it is the first word: there’s a lot of space in space.
Douglas Adams (in HHGTTG) is almost obligatory here: “Space is big. You just won’t believe how vastly, hugely, mind-bogglingly big it is. I mean, you may think it’s a long way down the road to the chemist’s, but that’s just peanuts to space.”
On top of LEO, even huger volume are available at MEO (perhaps most notably used by GNSS constellations like GPS and Galileo). In contrast, GEO — almost 36,000 km up — is fairly crowded in practice because the contention is over the nominal, equatorial positions.
Your reasoning is correct at the general level ie if you sprinkled satellites uniformly across the LEO volume. But they tend to cluster into thin orbital planes, so the actual spacing is way tighter (still not as dangerous as most people imagine though).