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For example on wikipedia for Switzerland it says the country has an area of 41,285 km². Does this take into account that a lot of that area is actually angled at a steep inclination, thus the actual surface area is in effect larger than what you would expect when looking onto a map in satellite view?
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.
It's the same problem with defining coastlines. You can keep increasing the resolution and the coastline length will increase indefinitely.
Ahh those fiddly little fjords.
those fiddly atoms and quarks
I would imagine that the area increases significantly, a type of example of what they say about fractal coastlines theoretically being able to have a perimeter of infinite length.
EDIT: it just occurred to me that theoretically, if measuring area with a different scale, a country like Bhutan could claim to have as much surface area as... say Australia.
Or both are infinite, but since one fits inside the other, I'm getting into that weird mathematical study of infinities within infinities.
The Gabriel's horn / painter's paradox is a good one too.
Never heard of it before, that's another mind melter. How does the volume of the horn end up to neatly be pi ?? YouTube link I found.
And what about navigation, does it count the slopes in? Is the route actually longer than it says if you travel up and down mountains?
It's probably aware of them, but generally no. Most slopes for driving on are smooth enough to be pretty negligible unless you're going hundreds of miles or more, in which case fives of miles won't make much difference either.
But if you're traveling by bike those small slopes may make some parts of the ride significantly more difficult or easier, and for cars may impact fuel efficiency in a way much more significant than just counting the extra distance traveled. So many navigation systems will still account for slope, even if they don't necessarily acknowledge the length of your path as precisely as you may have hoped
Not for distance, however when you start doing fuel calculations it gets counted in