TITLE: Definition of area
QUESTION [18 upvotes]: I am looking for an attractive, but rigorous definition of area;
say in Euclidean plane. Probably there is no short definition. It is OK to make it even longer, but can it be built from useful parts in a not boring way? Say in such a way that not all the students will sleep on the lecture? 
Comments. 

The real problem is to prove existence, the uniqueness is easy.
Using integral does not seem to be a good idea.
There is an approach where you write the formula for area and then proving its properties. I do not like it since it moves you to discrete geometry which is completely irrelevant and the ideas used nearly useless anywhere else (so no reason to learn this stuff). [See for example "Geometry: A Metric Approach with Models" by Millman and Parker.]
The method with measuring grid (cutting everything into small squares and counting) looks much better. One can consider this method as an introduction to integral. There is only one technical statement which has to be proved --- if you rotate square then its area does not change. The only problem is that it is not generalizable to say absolute plane or sphere...
One may define the area as a limit of $\varepsilon^2\cdot N_\varepsilon$, where $N_\varepsilon$ denotes the maximal number of points in the figure one distance $>\varepsilon$ from each other. The only hard part is to prove existence of the limit $\varepsilon^2\cdot N_\varepsilon$ for say polygons. (You can exchange the limit to ultralimit --- this way everything works smoothly, but I do not want to sell my soul just to get a def of area...)

REPLY [8 votes]: I don't know how much about areas you want to prove, and  how developed a background the audience is supposed to have,    but here is a definition of area of a bounded planar region. 
Take  such a region  $D$ in $\mathbb{R}^2$,  $\newcommand{\bZ}{\mathbb{Z}}$  $\newcommand{\ve}{{\varepsilon}}$ consider the $\varepsilon$-grid $(\varepsilon\bZ)^2$ and denote by $N_\ve(D)$ the number of $\ve$-pixels that touch $D$. (A pixel is one of the $\ve\times\ve$-tiny squarers of the grid.) We then declare $D$ to be measurable (i.e. to have area) if the limit $\newcommand{\eA}{\mathscr{A}}$
$$ \eA(D)= \lim_{\ve\to0}\ve^2 N_\ve(D) $$
exists.   If this is the case, then we  define the area to be the limit $\eA(D)$, and we set $\eA_\ve(D)=\ve^2N_\ve(D)$.  
The first step is to prove $\newcommand{\bR}{\mathbb{R}}$ that if $L,U: [a, b]\to \bR$ are Riemann  integrable functions  and  $D(U,L)$  is the region 
$$D_f= \bigl\lbrace\; (x,y)\in [a,b]\times \bR;\;\;L(x) \leq y\leq U(x)\;\bigr\rbrace, $$
then   $D(U,L)$ is measurable
$$\eA(D(U,L))=\int_a^b  \bigl(\; U(x)-L(x)\;\bigr) dx. $$
The next thing to prove is a weak form of the inclusion-exclusion principle: if $D_1$, $D_2$   are  measurable   regions   that intersect along the graph of a $C^1$-function, then $D_1\cup D_2$ is measurable  and 
$$\eA(D_1\cup D_2)=\eA(D_1)+\eA(D_2). $$

REPLY [7 votes]: To be honest, I'm no sure if this consrtuction matches the criterion of non-boringness, but it is rather short (I promise you all this takes less than 40 pages ;) ) and works in every dimension. It is related to the answer of Andy Putman, and the subsequent comments. I should maybe have kept on commenting, but there was simply not room enough. 
For my physics classes I introduce measuring the area of bounded open and closed sets (or volume in $\mathbb{R}^n$ ), by considering countable unions of closed rectangles, the edges of all of them being parallel to a given system of cartesian coordinates (let's call them acceptable). The union must be clean, in the sense that the rectangles overlap neatly to form a rectangulation (say). Because any nonempty intersection of two acceptable rectangles is again an acceptable rectangle, any countable union of acceptable rectangles clearly admits a clean sub-rectangulation.The same argument works also pretty well for describing a common sub-rectangulation of two others having the same image.
Define the area first for finite unions, starting by assigning the usual value to the area of a single rectangle (given as axiom), and extending the area functional using the usual additivity for quasi-disjoint (intersect at most along a common edge) union of acceptable rectangles (given as axiom). You retrieve all the usual properties of an area in that case.
Now a bounded countable and clean union $\bigcup_{n\in\mathbb{N}}R_n$ of acceptable rectangles can be assigned its area as the series $\sum_{n\in\mathbb{N}}\mathtt{Area}(R_n)$ (which always converges). As pointed out above, finding a common sub-rectangulation is fairly straightforward, and the limit does not depend on the choosen sub-rectangulation (commutatively summable series).
Since any open set is rectangulable (in the sense that it breaks in a countable clean union of acceptable rectangles), we can measure any open set. Any compact $K$ is included in the interior of an acceptable rectangle $R$ and $R\setminus{K}$ is a bounded rectangulable set $O$. We assign $\mathtt{Area}(R)-\mathtt{Area}(O)$ to the area of $K$.
By adding more natural axioms you can again extend this functional to all bounded elements of the borelian tribe, but that's a classical story.