Before satellites and other modern techniques, how the area of countries/territories/regions used to be calculated?
I can understand that plotting boundaries is an easier task, even without the advanced technologies we have today. However, finding the area is a much more complicated task because of uneven terrain, forests and buildings/structures etc.
Satellites and modern technologies are irrelevant here. If you have a map of the country, break it into sufficiently small pieces (so that each piece is approximately flat, and the scale of the map is approximately constant on each piece), then put a sufficiently fine square grid on each piece and count the squares.
Or use the simple device called a planimeter directly on the map of the small piece to find the area of the piece on the map. Then use the scale. (The mathematics on which the planimeter is based is Green's theorem. It was known in the middle of 19th century). Here is a nice photo showing geographers using planimeters.
Yet another clever method was used in 17th century (by Galilei, for example): carefully cut your country piece out of the map with scissors and weight it on a fine scales. Comparing with the weight of a square piece of the same paper, you obtain the area.
If you have a map of the country This seems incredibly unaccurate, because A) level of detail of the map and B) as projections on a plane of a sphere, all the maps suffer distortions.
As Alex answered, the math was well known, and the mapping problem could be solved by surveying techniques. i.e., by repeated measurements of angles and known distances, as you can see the students doing in any modern civil engineering course.
It was obviously not as exact as modern mapping. I remember seeing a globe from 1938. The shapes were very close to the correct ones, but still they were noticeably different even to the naked eye.
Manual surveying could be an herculean task. The Cassini family is well known by a massive 4 generation, 65 year effort to map France at the behalf of the king:
https://en.wikipedia.org/wiki/French_cartography#Cassini_maps
Here you can see the best known map before and after the Cassinis work:
https://upload.wikimedia.org/wikipedia/commons/6/6c/France_cotes_academie.png
It is easy to note how Cassini's map is much better than the previous one.
The mark zero was Paris. Triangulation error accumulates from Paris, therefore there are more errors on the south of France.
You can see their map in detail overlayed with modern maps with a google-earth interface here: http://rumsey.geogarage.com/maps/cassinige.html
It is worth to have a look, it is much more accurate than what you would think.
The Cassinis had lots of funny and tragic stories over their decades long work. Everybody knew that if Le Roi could have accurate maps, then His majesty's taxation would also be more accurate. Therefore, the surveyors had lots of enemies. Sometimes angry landowners would kick them out of their dear lands or even shot them, lest the king have an accurate measure of their riches.
