# How is the area of a country calculated?

As countries' or states' borders are not straight lines but they are irregular in nature, I wonder how anyone can calculate the area of a country or state.

When do you think the area of a country or state was first calculated? Was it before satellites provided us accurate picture of the earth?

Note: I am not asking about surface area of a country. They are assumed as flat while calculating the area.

• "They are assumed as flat"... I don't understand what that could possibly mean. Moreover, any measurement of the real world is approximate - there is no such thing as "the" area of a region of the Earth's surface. May 28 '13 at 4:14
• @ZevChonoles I mean any projections like mountains or lake or rivers, which alter surface area are ignored. and whole surface is considered as flat. May 28 '13 at 4:19
• I suggest you add that clarification to the question.
– Joe
May 25 '16 at 3:34

I guess you could ask Google or WolframAlpha. Interestingly, these answers differ substantially. Perhaps, that's just a matter of how territories are interpreted but it illustrates the point that, there's really no easy answer. The question is at once terribly elementary and, on the other hand, fabulously interesting. Mandelbrot asked the question "How long is the coast of Britain?" Turns out that it depends strongly on how carefully you measure it.

So, the short answer is - it's super simple, in that you do it just like any other spherical polygon. Dealing with data at this level, as well as territorial disputes, is a bit more complicated. To illustrate more concretely, consider the image below.

You can see that, in a very simple sense, each "country" can be triangulated. These triangles can then be further sub-divided to get a good approximation and the areas of the triangles can be added up. Thus, again, on the most basic level it's no harder than adding up the areas of some triangles. To fully implement this, though, requires a fair understanding of computational geometry as well as access to solid data, which is likely disputable anyway.

The real question is - can you get good upper and lower bounds?

Per request, here's a look at the other side of the planet:

And here's a look on the map of just India and its neighboring countries.

• Both of them Google and Wolfram has arrived at some kind of value. How did anyone calculated that value ? That is my question May 28 '13 at 4:28
• @VAR121 Does my edit make it at all more clear?? May 28 '13 at 4:43
• It is clear now. thanks for your help. May 28 '13 at 4:45
• Can you please upload the globe showing other side(Asia, esp India) of the globe. May 28 '13 at 5:04
• @HSN That is interesting, thank you. Initially, I was thinking that these sorts of triangulations are quite different. A bit of Googling about revealed this, which is in English. Those triangulations are pretty impressive. The triangulations presented here, though, are generated using the program triangle, which I'm pretty sure uses some new ideas. May 28 '13 at 12:02

One easy way to do this would be a Monte Carlo method. Say you had a map of a country with map area $A_{map}$. Throw a large number $N$ of small seeds randomly on the map. By the number of seeds to fall within the country $n$, one can find the area of the country: $$A_{country} = A_{map} \cdot \frac{n}{N}$$ This of course ignores maps with projections that are nor area preserving, but never the less.

• I guess projections should always(if you want to calculate area not surface area) be ignored. May 28 '13 at 5:28
• @VAR121 Some map projections, so-called equal-area projections, represent areas in their true proportion so you could use one of those. May 28 '13 at 5:31

Geographers used to measure areas with a planimeter:

http://en.wikipedia.org/wiki/Planimeter

Nowadays I suppose they use something like ArcGIS, which has a built-in area calculator, but I don't know what algorithm is used.

suppose that you have an area approximated map(maybe calculated by some complex optics from a satellite image) and as above said if area of map is Amap, spread some fine particles(sand or any) on the map evenly/uniformly accross, then somehow seperate out the sand covering the area of country, pour it into a glass test tube to measure its height, then compare that height with full sand height, that gives the more accurate ratio(n/N) or here (h/H) as above said Acountry=(h/H)Amap

• Welcome to stack exchange! please consider adding more details, and use MathJax to type equations. Jul 17 '19 at 19:32