Hi! First post on this particular branch of the site, though I've been hanging around the sister site for a while longer. This seems to be the place to come for actual answers.
Or Southern Hemiplane, depending where you put the centre at. Whichever way, on a Flat Earth it would be a circle with a circumference at, for example, the equator. Under regular FET, it's just part of a larger disk, but it's still a circle.
Of course, it doesn't need to be an exact circle, but a circle's the best-case scenario.
Now we get onto maths. I'm studying it, and there's something known as the 'Isoperimetric Inequality'. I can give you a proof of that if you really want, but it's fairly long, and pretty advanced: unless every mathematician is knowingly in on the conspiracy, however, it should be worth noting it has been around since the 19th century (rigorously proven), and known generally for much longer. You're more than welcome to look it up, there are several proofs online.
Essentially, it's a relationship between the boundary of a shape, and its volume: but we only need to worry about the two-dimensional case. That is, with a flat shape, with perimeter, and area.
The inequality provides a bound: the area of a shape can be no larger than a certain function of area. The inequality is:
L2 >= 4pi*A
Where L is the length of the perimeter, A is the area. Equality holds only with a circle: what this means is, as we know the length of the equator, we can calculate the largest possible size of the hemiplane contained within it on a flat Earth. Strictly speaking we just need to calculate the area of a circle with that circumference, but it's nicer to use the inequality form as it shows explicitly we're concerned with an upper bound.
The equator has length of approximately 40,000km. Plug that into the inequality, we find:
A <= 127323954.5km2
Or around 0.13 billion (the American kind).
Just for fun, we can also note that's also approximately the area of a circle with said circumference (rounding errors aside).
Anyway, that gives us the largest possible area for the inner hemiplane. For comparison's sake, that's little more than the area of Afro-Eurasia: one supercontinent which mostly takes up a third of the northern hemiplane. It's also about half the area of what we'd expect on a spherical Earth, after a hasty calculation (and while the Earth is not perfectly spherical, I rounded the FE number up and the RE number down). Even if there's some concave/vexity, that's one hell of an error bar.
So, can we blame this all on the conspiracy? Is every number we've been given wrong? Are you suggesting not one cartographer, pilot or anyone has noticed that the distances, calculated by Round Earth numbers, fall drastically short? If anyone wants to work out the speeds at which planes would have to be travelling under FET given the area of the necessary hemiplane, you're welcome to; might be too short to keep lift, I don't know offhand.
Or is all of maths just wrong?