How and when could a Dyson sphere civilization figure out the shape and size of their sphere?

Kinda inspired by the non-disappearing ship question, but I actually asked this particular question on an unrelated forum a few years ago. Reposting it here (well, paraphrasing, really, since I don't actually recall much of the original specifics) because I'm interested in the opinion of this particular community.

Consider a typical science-fiction Dyson sphere - a few hundred million miles (or, if you prefer, a few AU) across - with a human (or essentially human) civilization living on it. Many different such civilizations, actually (since the sphere is very big).
They've been there since an awful long time ago; long enough to have developed all of their technology beyond the early stone age while already on the sphere, and long enough that not even legends preserve any mention of the original arrival (I'm assuming, for clarity, that such an arrival did indeed happen, perhaps 100,000 or so years earlier, but it doesn't really matter).

With that in mind, at which point in tech level (assuming an advance roughly along an Earth-like technology tree) would they be able to figure out that they live on a giant sphere (as opposed to, say, a giant flat plane), and at which (presumably later) point would they be able to figure out the approximate size of their sphere? And how exactly (what sort of tools, methods, calculations...) would they be able to do that?
(Your choice of what counts as "approximate"; if you have several different scenarios for different levels of precision, I'd be happy to see all of them.)

There are really two different questions here (plus an addendum), for different definitions of "Dyson sphere", which I'm putting together because I'm not sure I could write enough detail for both separately. (I'd be happy to make a short separate question for one of the versions if the mods think that it would be better.)

1. A regular (for science fiction, at least) Dyson sphere, with an inner-facing habitable surface. Light and heat is probably provided by a star in the middle; gravity must be artificial, because the natural gravity balances itself. Your choice on how the day/night cycle works (or whether there even is one) - this might matter for the specifics, obviously.
2. An inverted Dyson sphere - a sphere of similar size (several hundred million miles, i.e. a few AU, in diameter) with a habitable outer surface. Will probably have stars orbiting around it for light and heat (might or might not, depending on the specifics, also work as a day/night cycle). Natural gravity is easy in this case - just make the sphere sufficiently thick (a few thousand kilometers); but in this case there's not much difference whether it's natural or artificial gravity.
3. The addendum version - same question (when and how will the inhabitants any other hyper-large space habitat (such as a Ringworld like the one in the Niven series, or a Culture-style orbital).

Just for the record - I'm looking for reasonably realistic technological solutions (aside from the existence of the sphere itself), not fantasy.
If you happen to think (I personally doubt it) that there's no real way to figure out the size (at least, within the current level of Earth technology) - say that, and explain why.

...If this question is off-topic, sorry. (You can also try to fix the tags, if I chose them incorrectly.) This is my first question on Worldbuilding SE (and I hadn't asked very many questions on other SE sites before either).

posted over 8 years ago

January First-of-May‭

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