TITLE: What is the formal, precise definition of area (and that of volume) in geometry?
QUESTION [14 upvotes]: Note: I am asking two questions. Replace "area" by "volume" with appropriate adjustments to get the second question
Hello everyone. I am studying mathematics in college and I am trying to find the formal definition of the notion "area" in geometry. What is area? I can understand it when the sides involved are integers, like when the side is 3, we have nine 1x1 grids, so the area is 9 if we think of area as the number of "unit grids". Maybe we can also define area for squares with rational sides in the same vein, but this is not a good way of thinking because how to define the area of a square of side PI or an irrational number for example? 
Please give me a formal definition of area. I am looking for the most precise definition. I am not finding any results on the internet. Thanks in advance

REPLY [5 votes]: The other answers so far have mentioned basically three kinds of unsigned area:

Scissors-congruence to compare areas. Note that this approach does not define area itself as a magnitude, unless you further add the notion of real numbers. It also fails for non-polygons unless you take limits, in which case it becomes equivalent to the next kind of area.
Jordan measure (equal to both the limit of the total area of grid squares contained in the figure as the grid size tends to zero and the limit of the total area of grid squares needed to contain the figure as the grid size tends to zero, if these limits are equal). Any bounded object in classical geometry has finite Jordan measure, and this is the simplest intuitive notion that the classical geometers would surely agree with, since the method of exhaustion is essentially based on the same principle. Jordan area defined is finitely-additive and hence classical geometry works as usual.
Lebesgue measure, which assigns a countably-additive area to some sets of the plane, and is compatible with Jordan measure whenever the Jordan measure is defined.

All of the above generalize easily to higher dimensions.
However, there is also signed area, which can drastically reduce the number of cases in Euclidean plane geometry when it comes to polygons. We define (signed) area of a polygon using the shoelace algorithm, and then check that $area(XP...QYR...S) = area(XP...QY) + area(YR...SX)$ (the area of a polygon is the sum of the areas of its parts), and that area is preserved when the vertices are cycled but negated when they are reversed. This allows us to handle most geometry problems without having unnecessary cases, unlike if we use unsigned area.
Nevertheless, it is not so easy to extend signed area to non-polygonal regions. We can define the (signed) area of a region bounded by a (directed) rectifiable closed curve as the limit of the area of the approximating polygon. It turns out that the limit exists, but is not so easy to prove. (To see that the condition of rectifiability cannot be dropped, note that an Osgood curve is a continuous closed curve with positive Lebesgue measure, and hence its interior is not Jordan measurable, which implies that the limit does not exist.)
Also, signed volume is even more troublesome to define, so it is not very useful compared to the Lebesgue measure for general sets in $\mathbb{R}^3$.