I can't remember if it was Matt Parker or Tom Scott who did a thing about this on youtube, and I think the answer was no, they just use whatever method is easiest to get these numbers, and they're not even using a consistent methodology from one country to the next.

No. It's a flat approximation. The short answer is that once you take account for topography, your answer will always grow with surface resolution, and thus the actual surface area of rough topography is undefined.

Inclination must be ignored for area to work on a map, plus the inclination ends up being lost in the noise on a large scale. It is very similar to the coastline issue where the more detail you include the longer the coastline gets until every coastline is basically infinite.

Let's take an area split into a grid. One are has a hilly round slope, one is flat, and the rest are a variety of combinations. If you tried to take slope into account the one with the round hill would require the straight lines of the grid to warp towards it like one of those space time curvature pictures. The one that is flat is the only one that could be square, and even then it only works if you count it as flat since even flat ground has a small texture.

So no, they don't take elevation into account for maps because it would be far too difficult to measure.

The surface area can only be calculated with a defined level of accuracy due to how textured surfaces work.

It's not, but I love the idea that it could be. You just know some of those megalomaniac dictators would be piling up fake Hills to make their country bigger. Turkmenistan would have giant Towers of dirt everywhere.

There‘s a great video on that topic by Matt Parker (Standupmaths) which I‘d wholeheartedly recommend.

TLDW: No, terrain elevation isn‘t generally taken into account when calculating a country‘s surface area, though the actual method differs slightly from country to country. Switzerland would be around 7% bigger, if its mountains were to be taken into account.

No.

It's actually worse than that. Very few borders are straight lines. We have to approximate the border when calculating land area.

In addition to the other reasons people have given for why they don't: we're usually less interested in the surface area of a plot of land than we are in the usable area.

People generally can't live or farm on a significant slant and will instead level the ground or build supporting structures to make things level.
Things like rainfall are also generally better calculated by flat area.

Doubt it. Because then it becomes a question of resolution, where too high results in countries with infinite area, since ever slope, ditch, and grain of sand has to be accounted for.

BTW are the walls of New York's scyscrapers included in the calculation?

/s

Switzerland mentioned 🇨🇭🇨🇭💪💪

No, what for? Most of it is unusable anyway. And the less steep inclinations are in the margin of errors.

I don't know the precise answer, but I do know this:

1. Often the kind of measures that are about something vast and complex (like population for example) are really good approximations, not completely exact numbers. So maybe doesn't matter because the number itself is not trying to be 100% accurate.
2. As far as I know those measures are made from the top down view, like with airplanes or satellites, so no it would not include inclines. To include inclines in a precise way it would have to be measured each one on the spot, which is not the way that is done. There are almost no field surveyors these days, again, as far as I know. And to include inclines in an approximate way takes us to point 1 again so it wouldn't matter much if there were a small difference.
3. Why would we do that? Almost everything we use land for requires it to be horizontally flat, so we flatten it. For example, an irregular coastline doesn't matter because we can use the crevices and irregularities to fit in more boats or ports or beaches, since the sea is horizontally flat and that is what really matter to us. But if there's a hill with a greater area because of the steepness of it we cannot fit more houses or warehouses of streets. We have to flatten it first so we gain noting from it being inclined.

So form my point of view it would be almost as if we tried to include the sides of a ravine or gorge in the measure just because technically it is area space.

Sorry if I cannot give you an exact answer, but I wanted to comment because you raised an interesting point that made me think.

Cheers.