Plane actually has a lot of definitions that seem different but are all related. It’s a flat surface, it’s short for airplane, and a bunch of definitions that I didn’t know of like a tool for making things smooth, a verb form that means to make smooth, and a tree of the Platanus genus (yes, it really is related to the others). And also there’s one that means to glide, which, I’m calling it right now, is where airplane comes from.
The plane we still use, a flat surface, showed up in the early seventeenth century, although the verb form to make smooth is from the early fourteenth century. It comes from the classical Latin planum, which means, you know, plane. Not trying to be original here. Anyway, before that it was the Proto Indo European pla-no-, which is a form of the word pele-, flat or to spread out. Funny how the only one we really still use is the one that’s most recent. Only three hundred more years and it will be as old as the others before they fell out of use.
Most of the other planes are related to planus in some way. The tool one and the make smooth one are pretty obvious, although they actually came to us via the Old French plane/planer. Airplane of course is just air + plane (told ya), and it’s the same story for aeroplane. But there are plenty of other words that are related to plane.
Planet comes from the Old English planet. Which means planet. It comes from the Old French planete and Late Latin planeta, and earlier the Greek planetes, which of course just happens to be from pe-le-. Maybe planet comes from flat because they used to think planet was flat.
Plan is also related. And also surprisingly recent as it showed up in the late seventeenth century as a noun and not a verb until 1728. It comes from the Modern French plan, you all know what this means, it’s plan. Look, there’s just no originality here. It comes from the classical Latin planum, which I assume you remember from a few paragraphs ago. As for why it went from a flat surface to a plan, well… what do you make your plans on? I’m not kidding here. Think about it. Have you ever tried making a plan on a jagged surface? It’s impossible!
Tony Jebson’s page on the Origins of Old English