A new way of thinking about $\pi$

Question

I don't Know, I should ask this question or not, But I am asking this -

(Before anything, this is an analysis of closed 2D figures.)

I was analyzing $\pi$ . It seems to be a constant, But being a constant is my curiosity. Can we say that $\pi$ is a property of circle which makes circle a circle. Same as there must be constant for each and every figure possible. So I suggested a new theory of $\pi$ or a new definition of $\pi$, which says -

$Figure_\pi = {L \over D}$

where, L = Length of the curve,

D = Maximum distance possible between any two points

So, $Circle_\pi = {Circumference \over Diameter} \implies Circle_\pi = \pi$

Also, $Square_\pi = {Perimeter \over Diagonal } \implies Square_\pi = {4a \over a\sqrt2} = {2\sqrt2}$

And, $Rectangle_\pi = {2(a+b) \over \sqrt{a^2+b^2}}$

Now I got stuck, as for rectangle I am not getting a constant, But I got to know, this is obvious. Two squares or circles can be similar but two rectangles can't be similar like that because for two rectangles to be similar, their ratios of lengths and breadths must be constant. So I suggested types of figure on the basis of their variables.

As a circle or a square can be defined by a single variable so I called them single variable figure. For rectangle or rhombus or ellipse we must have two variables, So I called them 2 variable figures. Or for any random figure to be similar we must know by some means the length of the curve and maximum distance between any two points and we will get a constant calling it $Figure_\pi$

My question is that, Am I thinking in right direction or not, because I am finding different constants for different 2D figures as different variable figures.

Answer 1

For any given shape you can define the ratio between the circumference and the longest chord, which will be a constant regardless of scale. This is because when you double the size of the shape you double the circumference and double the chord, so the ratio stays the same. You are correct that there is not a single ratio for all rectangles, but there is one for rectangles of a given proportion. Your $Rectangle_\pi$ can be expressed as $$\frac {2+2\frac ba}{\sqrt{1+(\frac ba)^2}}$$ where the dependence on the proportions of the rectangle is explicit. For a regular hexagon, the constant is 3. It will approach $\pi$ as the number of sides increases. You might see Wikipedia under Polygon approximation era for the story of Archimedes calculating $\pi$ in exactly this way.