Axiomatic geometry - How do you define the measurement of lengths, areas, angles, etc.? In an axiomatic approach to geometry (i.e., excluding explicit construction from $\mathbb{R}^2$), what is the best way to define numerical concepts, like length, area or angle measuring? 
I've searched on the Internet but I didn't get a satisfying answer to this question. For example, in the formulation of the Pythagorean Theorem you need to define length, and also square/sum of lengths. Do we have to enter in real number theory for that or is there an easier way to do that?
 A: This answer describes the way to introduce measures of segments and angles within Hilbert axioms of neutral geometry (i.e the theory which excludes Euclid's fifth postulate as well as its negation and is a basis both for Euclidian geometry and hyperbolic geometry).
First of all, among primitive notions we have ternary relation $B$. Let
$$B(abc) :\iff (a,b,c)\in B$$
$B(abc)$ means: point $b$ lies between points $a$ and $c$ (on the same line).
Secondly, by means of this notion we can define halflines, angles (pairs of halflines with the same origin, different and not complementary) and ternary relation $D$ between halflines as follows:
$D(ABC)$ if and only if halflines $A,B,C$ have the same origin $o$, there exists a line which passes through $A,B,C$ and not through $o$ and for every such line $B(abc)$, where $a,b,c$ are common points of this line and $A,B,C$ respectively.
Another primitive notions are congruence relations $\equiv$. (They are defined on different sets but we use the same symbol).
One of them is a binary relation between segments (i.e sets of the form $\{a,b\}$ where $a\neq b$). We write $ab\equiv cd$ to denote $\{a,b\}\equiv\{c,d\}$
The second one is a binary relation between angles. We write $AB\equiv CD$ to denote $\{A,B\}\equiv\{C,D\}$ ($A,B,C,D$ are halflines).
Let $\mu$ be a function defined on the set of segments with real values. We call it a measure of segments iff
1) $\mu(ab)>0$ for every segment $ab$.
2) If $ab\equiv cd$, then $\mu(ab)=\mu(cd)$. (Conguent segments have equal measures).
3) If $B(abc)$, then $\mu(ab)+\mu(bc)=\mu(ac)$. (Additivity).
Measure of angles is defined analogously.
It can be proved that both segment and angle measures exist. Moreover, for every measures $\mu,\mu_1$ there exists $\lambda>0$ such that $\mu_1=\lambda\mu$.
