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.
 A: 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.
A: 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.
A: 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?
Addendum
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.

A: 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
