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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, what for? Most of it is unusable anyway. And the less steep inclinations are in the margin of errors.
No.
It's actually worse than that. Very few borders are straight lines. We have to approximate the border when calculating land area.
Switzerland mentioned 🇨🇭🇨🇭💪💪
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.
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?
Not for distance, however when you start doing fuel calculations it gets counted in
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
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.
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
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.
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.
But here we have Switzerland as the example :) where nearly every small or large piece of farmland is far from level.
"significant" being the keyword. :)
There's clearly a slant, but you can see where it was a bit too much and so they added retaining walls to level things out.
In a place like this, you can tell what places are suitable for farming, which could be farmed with a little leveling, which are suitable for grazing, and which are just too steep for food.
They're definitely not going to entirely level every place, but you also can't grow food on the side of a mountain. :)
There's clearly a slant, but you can see where it was a bit too much and so they added retaining walls to level things out.
I have seen farmers work on steeper slopes (not saying it would be easy...).
I would rather guess the purpose of these retaining walls is to slow down the soil erosion.
even the little bits of grass around the peaks in foreground could actually be used. I'm amazed how risky cows behaviour is regarding to the abyss, and goats somehow are just completely not afraid of heights at all and hop around on 400 meter cliffs like a walk in the park. So you can grow food (meat+milk) on mountain sides during summer
Yup, it's why grazers are so common in mountainous areas. It's way easier to manage goats and cows in a mountain, with some fruit, hay and wheat fields to supplement down in the valleys than it is to reshape the steep areas to make them suitable for crops.
It's why Switzerland has it's own type of cheese, and the flattest parts of the US are predominantly known for "lots of corn".
To be clear though, the unusable areas I referred to were more the mountains in the background, or the nearly shear cliffs in the middle ground that the shepherd is unlikely to let the (perfectly willing) goats graze on, on account of needing to be able to get the goats later. :)
The photo of the terraced farming actually brings up an interesting point--in order to render those slopes usable for farming, terracing approximates the "flat" projection of the terrain anyways, so you end up with the same result. Buildings and any other usable structures follow the same rule: you can only build vertically, so the effective surface area is the same as the flat projection.
BTW are the walls of New York's scyscrapers included in the calculation?
/s
Ok, so I know this is just a joke, but you got me thinking.
Each floor of a skyscaper with an area of x² ft, is an additional x² ft of unique space. An acre of skyscrapers, could have 20-30 times more useable square footage than an acre of farmland.
You just invented the arcology again, lol
It is a near future concept of a fully self sufficient megacity, a sort of intermediate stage towards an ecumenopolis (a planet entirely urbanized with an extreme population density). One popular example would be coruscant, the capital world of the galaxy in star wars.
So, what do you think: has a skyscraper more wall area (including the inner walls) or more floor area (counting all floors)?
That's a fun puzzle like "how many golf balls could you fit in a 747", but to keep in line with the spirit of the original question, I think floor space is more in line with "is the area of a space more complicated than it's 2D outline as seen from space" than walls are.
Does that sentence make any sense rereading it? It seems very confusing.
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.
This is the same problem as with coastline length - the higher resolution you'll use, the bigger the number you'll get. AFAIK this is solved by just standarization of calculation methods - there's just certain rounding to be applied in all cases so you get a comparable results