Proof that every circle has the ratio of $\pi$ We know that the ratio between the circumference's length and diameter is equal to $\pi$, but can this be proved for every circle? Or is this an axiom?
 A: It is somewhat of a delicate matter. Firstly, the independence of the circumference-diameter ratio from the radius of the circle is not true in all geometries. It is in fact a characteristic of 'flat' geometries, or, to use the standard term, of Euclidean geometries. It is quite easy to see for instance that this ratio is not a constant for circles on a sphere. Also, using any of a number of models for the hyperbolic plane, the ratio in the hyperbolic plane is also not constant. 
Next, the notion of length of a curve is very subtle, and requires a good definition. The issue with length is that it is very sensitive to small changes, and is not continuous (in the sense that for curves that uniformly converge to a given curve, the lengths of the curves need not converge to the length of the limiting curve). This makes approximating a length of a curve by geometric means subtle and error-prone. 
One solid way to proceed is to accept the definition of the length of the graph of a differentiable function $f:[a,b]\to \mathbb R$ to be given by $\int _a^b\sqrt{1+f'(x)^2}dx$. This definition is obtained by using the Pythagorean theorem, which has hundreds of proofs. Now, the upper semi-circle of radius $r$ is the graph of the function $f:[-1,1]\to \mathbb R$ given by $f(x)=r\sqrt{1-x^2}$. Plug that into the definition of length and you'll get $2r\pi $. Thus the circumference is linear in the radius, which is why upon division by the diameter, you always get a constant. (To make this argument more rigorous, one also needs to prove that length is translation invariant, so that any circle can be translated to have its centre in the origin).  
Remark: The ancient greeks argued differently about the constant $\pi$, since they did not have our modern definition of length. I'm not sure how aware they were of the subtleties of the length notion. Arguably, it is better to first consider area and not length. $\pi$ can then be defined as the ratio of the area of a circle of radius $r$ to the square of the radius. The proof of this ratio being independent of the radius uses integrals again, but to measure the area under the graph, rather than the length of the graph. 
A: Here's what Euclid had to say on the matter. Ostensibly, an equivalent sort of argument can be made as follows:
We note that for any two circles, an inscribed square has perimeter proportional to the diameter of the circle.  Similarly, we can show (inductively?) that an inscribed regular $2^{n}$-gon has perimeter proportional to the diameter of the circle. 
We then may state that the sequence of lengths of successive inscribed regular $2^n$-gons of a given circle forms a monotonically increasing sequence that bounded above, since the length of an inscribed regular polygon is less than that of the circumscribed polygon (of the same number of sides). 
Thus, there exists some limit of the lengths of these $2^n$-gons, and we may "reasonably" take this length to be the circumference of our circle. Since each $2^n$-gon has length proportional to the diameter of the circle, we may state that the circumference of the circle must also be proportional to its diameter.  Thus, the ratio of a circle's circumference to its diameter must be constant.
A: It can be proved that the ratio $\frac{\text{circumference}}{\text{diameter}}$ is the same for all circles, no matter how large or small; in other words, this ratio is a constant number. The symbol $\pi$ is used to denote this constant number. We need a special symbol because there's no other good way to write the number, really, since it is transcendental. You can easily write down approximations like $\tfrac{22}{7}$ or 3.14159265, but these are only roughly equal to $\pi$, not exactly $\pi$.
In short, then, $\pi$ is defined to be the ratio $\frac{\text{circumference}}{\text{diameter}}$. Using your term, this is an "axiom".
I'm guessing that your knowledge of mathematics is fairly elementary, so I gave you an elementary answer that ignores some subtle issues. For example, I assumed that we're working with standard planar Euclidean geometry, and that we have a workable intuitive notion of what "length" means. 
