How is area defined? Thinking about area in the context of the Lebesgue measure, I have an intuitive understanding of how area is constructed in $\mathbb{R}^2$:


*

*define all rectangles to have the area $length \times width$,

*the area of a shape in $\mathbb{R}^2$ is the smallest area obtained by covering the shape with rectangles.


In practice, when calculating the area of Great Britain, say, I have following picture in mind.

This process can be made rigorous via measure theory. Indeed it is the unique measure, if rectangles have area $length \times width$ by Carathéodory’s extension theorem.
However this process is limited to area in the plane. How do I think about the surface area of a sphere for example? And can I show that it is consitent with the notion above?
Is there a similar process for finding the surface area of any shape? If not, then how is this done?
 A: Carathéodory [1] provided a method to construct a $p$-dimensional measure in $n$-space that generalizes the standard $p$-dimensional content.
[1] C. Carathéodory, "Über das lineare Mass von Punktmengen, eine Verallgemeinerung des Längenbegriffs" Nachr. Gesell. Wiss. Göttingen (1914) pp. 404–426.
{Plug} English translation in:
[2] Classics on Fractals, Edited by Gerald A. Edgar. Westview Press, 2004. ISBN: 0-8133-4153-1
Summary for surface area in $3$-space 
Let $E \subseteq \mathbb R^3$ be a set. And we want the "area" of $E$.  For a positive number $\epsilon$, let
$$
\mathcal H^2_\epsilon(E) = \inf \sum_{k=1}^\infty \big(\mathrm{diam}(C_k)\big)^2
$$
where $\mathrm(C_k)$ is the diameter of the set $C_k$, and the infimum is taken over all countable covers of $E$ by sets with diameter at most $\epsilon$:
$$
E \subseteq \bigcup_{k=1}^\infty C_k,\qquad \mathrm{diam}(C_k) \le \epsilon
$$
and then let
$$
\mathcal H^2(E) = \lim_{\epsilon \to 0} \mathcal H^2_\epsilon(E)
$$
Adjusting by an appropriate constant factor (I think $\pi/4$) so that this agrees with the usual area for plane sets, we define
$$
\frac{\pi}{4}\;\mathcal H^2(E)
$$
to be the "area" of $E$.
A: Things are less obvious for curved surfaces. You can still consider a coverage by elementary shapes, but this raises a difficulty: even if you find tiling shapes (could be rectangle in a curvilinear coordinate system), you still need to know the areas of these shapes.
For a sphere, you can use the system of meridans and parallels, but it is a non-trivial matter to establish that the area of a tile is proportional to the sine of the colatitude.
