Do we need measure theory to prove length of diameter and circumference is constant is a circle?

Question

In high school I was taught that $\pi$ is the ratio of the length of circumference and diagonal of a circle. But is it necessary to use some measure theory machinery to define the length of circumference of a circle? Or length of a diameter? And that in a given circle, do I need measure theory to prove that all diameters have equal lengths or can the invariance of length in rotation proved some elementary way? I tried to prove that $\pi$ is constant so I wondered can I avoid using measure theory if I prove everything starting from the Hilbert axioms.

Answer 1

I presume where you wrote "diagonal", you meant "diameter".

Lengths of line segments are distances between the endpoints, and how or whether those are defined depends on what approach you take to foundations of geometry.

The length of a curve, such as a circle, can be defined as the smallest upper bound of all lengths of polygonal paths along the curve.

Say a curve starts at a point $p_1$ (on the circle, pick any point to be the starting point) and you move along an arc to another point $p_2$ on the curve, and then keep moving in the same direction along the curve to $p_3$, and so on until you reach the other end of the curve, which you label $p_n$ (on the circle, you'd have returned to the starting point). Then $$ \text{distance from $p_1$ to $p_2$} + \text{distance from $p_2$ to $p_3$} + \text{distance from $p_3$ to $p_4$} + \cdots\cdots \\ \cdots\cdots+ \text{distance from $p_{n-1}$ to $p_n$} $$ is less than or equal to the length of the curve. But now suppose you add some point between $p_1$ and $p_2$, and another between $p_2$ and $p_3$, and so on, and look at the sum you get then. Generally it will be a bigger number. But it is still no bigger than the length of the curve.

The length of the curve is greater than or equal to the lengths of all such "polygonal paths". But the length of the curve is the smallest number that is greater than or equal to the lengths of all such polygonal paths.

That is how the circumference is defined.