WolframAlpha data dump:

  • US total area: 9 631 000 km^2
  • US arable land: 1 650 000 km^2
  • US population: 319 000 000 people

How many km^2 of arable land would Kingdom of 10 million people require to support it's living requirements?

The world bank data file says 0.51 hectares per person for the US so times that by 10 million and convert to km2

But Americans (and Canadians) eat a lot of beef which requires much more land per pound of protein to produce than other forms of protein so that skews the numbers. Also, what is the definition of "living requirements"? Is it bare minimum to keep alive or comfortably alive? Are we talking poverty level subsistence or middle class?

The book "Stank On Zanzibar" starts with (among other very VERY weird shit) the human race, standing shoulder to should, could fit on Zanzibar and ends with ... a list of deaths then "Despite the foregoing, the human race by tens of thousands would be knee deep in the water around Zanzibar"

It's impossible to answer that question without knowing the level of technology available to those people as well as a lot of other things.
A nomadic hunter-gatherer society needs a lot of room per person, a society employing industrial scale agriculture with high yield crops, field rotation, efficient water recycling systems, intensive animal husbandry, etc. etc. needs far less.

That's one big reason why the current world population has more food per person available (on average) despite being far larger than what was estimated would lead to perpetual famine just a few decades ago (another reason is that the people coming up with the estimate were deliberately alarmist, their very agenda being one of forced depopulation of the world down to a few hundred million, they're still at it to this day).

Prior to the Green Revolution there was serious alarm about global famine, and most of the current alarm surrounds the unsustainability of the techniques and technology that was introduced at that time:
Fossil fuels used to power the farm equipment and produce the pesticides & fertilizers are becoming harder and more expensive to obtain, some argue we may have already reached or are about to reach peak oil.
Phosporus mines needed for the production of artifical fertilizers are starting to run out.
Water levels in aquifers used to irrigate farms are dropping at alarming rates and many rivers are only a trickle by the time they reach the ocean.
Similarly glaciers which feed the major rivers in asia are rapidly shrinking.
As well as increasing reports of resistance to the most popular pesticides evolving and spreading in the pest populations.

Hopefully, science & technology will solve some of these problems but generally research funds for agricultural science have been cut back by many gov'ts since the Green Revolution and cultural rejection of the latest technology (GMOs) is hobbling this possibility.

it's not cultural rejection, it's political rejection. It stems from the same reason as the call for "population control", the wish to massively reduce the population of the planet.
And what better way to do that than to cause famine, and prevent diseases from being cured...
That's also why DDT (which is perfectly safe) had to go.

Wow there is just so much wrong with that statement.
1) a decreasing population is considered bad for the economy (hence why western countries where the birth rate is below replacement have lenient immigration policies so the total population is still growing).
2) evidence shows the best way to decrease the population is by educating women and providing access to birth control. In fact, because of growth in this area the global population growth rate has already slowed by a lot -> the population is expected to stabilize at 9-10 billion within the next century.
3) Disease is generally too slow (cancer, diabetes, etc... don't kill until people's 50s or older well after they have reproduced). The only exception is infectious disease the treatment & containment of which continues to have broad support. The rejection of vaccines also doesn't and won't put much of a dent in the population because modern medical treatment can save almost all victims of those diseases and most don't have a particularly high kill rate even without treatment (<20%) really the only consequence would be an increase in disabilities.
4) DDT is fairly safe for humans but is not safe for the environment. The Great Lakes still have a high enough concentration of DDT to prevent any Bald eagle population from being sustainable (too few eggs survive to hatching for a pair of eagles to produce 2 chicks which will survive to adulthood). True we probably didn't need to ban it completely, but it was to toxic to be used for agriculture.

If there really was a master plan to drastically reduce the human population it would be done by creating an infectious disease which causes sterility (Or alternatively by releasing the ferret-ferret transmissible H5N1).

@ReverendJim If I gave you and your 9 friends farm. How many km^2 does it need to be so you can live out of it without starving? "basic living requirements by food", meant, enough to have nobody who starves, enough to give all pregant women food enough so that kid come healthy out etc. .
@GrimJack What the heck are YOU talking about!?

If I gave you and your 9 friends farm. How many km^2 does it need to be so you can live out of it without starving?

Where is the farm? is it prime arable land with loomy soil or semi-arid marginal land or is it waterlogged and only suitable for growing rice? How long is the growing season is it long enough for two crops/year or only one? Do they have access to buy farm equipment, pesticides, fertilizer, farm animals or only their bare hands?

Prime arable land. 2 crops/year. Jungle-like (selfmade) tools and animals.

As has been pointed out, there are too many variables to be able to answer that question. Aside from questions about quality of land, length of growing season, etc., there is the question about whether each family/group is expected to be self sustaining. I think we can discount pesticides and fertilizer. Arable land when properly maintained using crop rotation and fallowing does not require fertilizer. My grandfather was a market gardener as were all four of his sons so I have a little knowledge about this. None of them used fertilizer or pesticides. However, none of their farms could be considered self sustaining - all raised vegetables, no animals other than the horses my grandfather used for labour.

Another question is whether or not that land is to be worked by hand/animal power or by powered machines. My grandfather used machines pulled by horses. My father had a tractor and other powered machines.

I think the question does not have a simple answer, and a more complex answer that takes into account all the variables would require time and resources beyond anyone here. Having said that, I think someone with experience in agriculture and nutririon might hazard a guess as to the maximum number of people this planet could theoretically support under ideal conditions.

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Did you expect a specific value as an answer?

42 is as good as any. Sigh.

@ReverendJim I do realize that. I need approx. amount. Are we speaking of 1, 5, 10, 20, 40, 80, 200, 800, 5000km^2? I don't need really specific answers, just 'round it.

Well, from the world-bank census's, the empirical number today seems to put most "normal" countries (i.e., excluding island states and really large countries like Canada, Russia, etc.) between 0.1 and 0.5 hectares per capita, which is about 0.001 to 0.005 km2 or 1 to 5 thousand square meters. So, that's basically a patch of land between 30x30 meters to 70x70 meters. Also note that many countries on the higher end of the spectrum export and/or spoil a lot of their food, while countries on the lower end import food, ration it more, and/or starve.

If you look at the world data, you have about 7 billion people and about 14 million km2 of arable land, giving a similar ball-park figure of about 2000 square meters (about 45x45 meters) per person.

But, of course, there are many caveats to these figures.

For one, it doesn't take into account sources of food like fishing, hunting, raising animals (needs land for grazing, and that land is not considered "arable" because you're not planting crops on it), or sea-based cultures ("fish farming"). In other words, it is definitely not a measure of "how much land would I need to survive".

Also, modern agricultural practices (i.e., the "green revolution") is largely responsible for most of the dramatic reduction of these figures over the years. In other words, even if you had a pretty good estimate of how much land / resources is needed to feed one person, within our modern agriculture system, it would not mean that you could theoretically purchase such a lot of land and be self-sustaining with it, especially not if you regress towards methods that are "traditional", "natural", "organic" or "<insert favorite BS buzzword here>". Because the lack of modern technology and being a small operation would result in significant loss of production, and you would never be able to match up.

About the green revolution, consider that the amount of land being used today for food production (of all kinds) has remained pretty much the same for the past 60 years (and has even decreased a little, by some measures), while the world population has nearly tripled (from 2.5 to 7.2 billions). And I've heard that current food production could feed around 10-12 billion people (btw, most starvation / famine problems today are caused by improper distribution, that is, wealth distribution). Which makes sense since the green revolution has roughly quadrupled the yield of most food sources over the past 60 years or so.

You could also consider the total area of the Earth as an upper-bound, that is, it will consider everything, including places that are not used at all (deserts, high-sea, etc.). That number is also easy to computer, just divide the total surface (510 Mkm2) by the population (7B), giving you about 70,000 square meters (about 260m by 260m).

But on top of that, you also have to factor in all the things that are not sustainable or renewable. Using non-renewable resources either to power machinery, or to make fertilizers, pesticides and herbicides, or when using exploiting the food resource in a way that is not sustainable (such as depleting the source over time, such as over-fishing), then you cannot consider that as being "an amount of land / resources to sustain a human being".

In other words, this is a very difficult problem to work out. And there are entire research groups that work towards getting a better understanding of all these factors. But if you just want rough ball-park figures, there they are, i.e., a few thousand square meters (but 60 years ago, it would have been 5 times that, and the further back in time you go, the worse it gets).

Prime arable land. 2 crops/year. Jungle-like (selfmade) tools and animals.

Well during the colonization of the Americas a family farm was 65 ha. Those farms would have been a bit bigger than the minimum but they also had access the forged metal equipment (pumps, plows, etc...) which wouldn't be available in your hypothetical situation. There is very little information available for pre-metal civilizations so it's hard to know how big their farms were and pre-metalworking cultures typically use a combination of farming, hunting, fishing and foraging because without metal plows it's probably not possible to grow enough food to support yourself.

So I'd estimate 50-100 ha assuming the community is big enough to support the basic metalworking.

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I think the infrasctructure for effective distribution should also be factored in. Rail, road, shipping etc and all the forges, quarries, refineries, ports. Of course this is in addition to things already pointed out by others with regard to the manufacturing of chemicals and materials essential for the running of the agri-home.

Prime arable land. 2 crops/year. Jungle-like (selfmade) tools and animals.

IOW subsistance farming, slash and burn farming techniques. The least effective at all, and a prime cause for destruction of rain forests all over the tropic, a big cause for the arification of Africa and suspected of being part responsible for the arification of Australia.

It's back breaking work too. You'd best have some slaves or expect to either starve to death when you grow much over 40 years old because your back gives out or have a lot of children (most of whom will die in infancy) to carry on the work and hope they'll feed you.