Area in axiomatic geometry
Let's say we have axiomatic geometry as defined by Hilbert's axioms. For line segments, angles, triangles, squares, etc. we have the notion of congruency to determine whether two of them are "the same".
But this doesn't seem sufficient to determine whether two figures of different shape have the same area. For example, I don't see how the Pythagorean theorem can be proved using only the notion of congruency.
So basically my question is: how is area defined in axiomatic geometry?
Solution 1:
From Marvin Jay Greenberg's excellent "Euclidean and Non-Euclidean Geometries" ...
What does "area" mean [...]? We can certainly say intuitively that it is a way of assigning to every triangle a certain positive number called its area, and we want this area function to have the following properties:
Invariance under congruence. Congruent triangle have the same area.
Additivity. If a triangle $T$ is split into two triangles $T_1$ and $T_2$ by a segment joining a vertex to a point on the opposite side, then the area of $T$ is the sum of the areas of $T_1$ and $T_2$.
Having defined area, we then ask how it is calculated. [...]
Basically, any strategy for assigning values that satisfy (1) and (2) above can be reasonably interpreted as "area" in a geometry. The calculations are what make things interesting.
In Euclidean geometry, one derives that the "one-half base-times-height" formula satisfies the necessary conditions. (Note: So does "one-half base-times-height-times-an-arbitrary-positive-constant".)
In spherical geometry, one can show that angular excess ---that is, "angle sum, minus $\pi$"--- works as a triangle's area function (up to an arbitrary constant multiplier). (We can do a sanity check with a simple example: A triangle with a vertex at a sphere's North Pole, and with opposite side falling on 1/4 of the Equator, covers one-eighth of that surface of the sphere; therefore, it has area $\frac{1}{8}\cdot 4\pi r^2 = \frac{\pi}{2}r^2$. On the other hand, such a triangle's angular excess is $\left(\frac{\pi}{2}+\frac{\pi}{2}+\frac{\pi}{2}\right) - \pi = \frac{\pi}{2}$, which is, in fact, proportional to the calculated area. (If we work on the unit sphere, we get to ignore the constant of proportionality.))
In hyperbolic geometry, angular defect ---"$\pi$, minus angle sum"--- is the go-to function. (This is harder to check than in the spherical case, so I'll note a fascinating consequence: a triangle with three infinitely-long sides happens to have three angles of measure $0$; therefore, such a triangle's area is finite ... specifically: $\pi$! (Constant of proportionality ignored, to maximize the impact of that statement.))
So, because area calculations are so very different in these contexts, you can't expect a single formula to fall out of the basic axioms. At some point, you observe a phenomenon that satisfies (1) and (2), you declare "this is area", and you go on from there.
Importantly, there need not be any direct connection between a geometry's notion of area and that geometry's incarnation of the Pythagorean Theorem. The fact that squares erected on the legs of a Euclidean right triangle have total area equal to that of the square erected upon the hypotenuse is a neat "coincidence". This doesn't happen in non-Euclidean (spherical or hyperbolic) geometry: these spaces don't even allow squares!
(See Wikipedia for a discussion of the non-Euclidean counterparts of the Pythagorean Theorem.)
I've been a bit informal here, but hopefully I've shown that your question "How is area defined in axiomatic geometry?" is actually quite deep.