Different ways to define distance and the definition of $\pi$ I saw a post a couple of posts about "proving that $\pi$ = $4$", and I thought of using a naive way to say that it is not impossible.
We know that $\pi$ is commonly  defined like this:
$$\pi = \frac{\text{Circumference}}{\text{Diameter}}.$$
How will you measure the circumference? That is equivalent to saying that "What metric I will choose to measure the length of this curved line?" The important point is that the metric we choose will affect what value of $\pi$ we will get.For example, we will get $\pi = 4$ if we choose the taxicab metric $(|\Delta x|+|\Delta y|)$ and we will get $\pi = 3.1415...$ if we choose the Euclidean metric $(\sqrt{\Delta x^2 + \Delta y^2})$.
In short, we will get different answers based on the underlying assumptions we are taking beforehand. Is my approach wrong?
 A: I'd say that in the end it simply depends on how you define $\pi$. Most people (especially non-mathematicians) know the definition you mention, i.e. $\pi$ is the ratio between the circumference and the diameter of a unit circle. Implicitly we use the euclidean metric to define what a unit circle is (because who wouldn't think of the euclidean metric in everyday life first, right?).
Now I think what you are doing is exactly the 'easiest' way to justify an assessment like $\pi = 4$. But we'll have to deviate from the 'everyday life definition' above in the following way. Let's take a different metric, e.g. $d(x,y) = \max(|x|,|y|)$. What is the unit circle $S$ in this metric? It's
$$S = \{(x,y)\in \mathbb{R}^2 | d(x,y) = \max(|x|,|y|) = 1\} $$
You can easily draw this circle, and you will see that it turns out to be the square of length 2 centered around the origin, which has of course circumference 8. If we now apply our definition of $\pi$ being equal to the circumference divided by the diameter, we get $\pi = \frac{8}{2} = 4$. (Small remark: you get a different result when using the taxicab metric as you proposed. You can also draw the unit circle with respect to the taxicab metric to figure it out!).
So as often: It depends on your definition.
I hope this helps!
