In IV-1, Example 3 of Advanced Calculus of Several Variables, by C. E. Edwards Jr. the reader is told "Since $\pi$ is by definition the area of the unit circle,...".
The example provides a proof that $A=\pi r^2$, so that result cannot be used to justify the (unconventional) definition.
Every definition of $\pi$ I have seen in mathematical literature amounts to:
Given a circle of diameter $D$ and circumference $C$, the real number $\pi$ is defined as $\pi=\frac{C}{D}$.
I could propose a number of alternative ways of defining $\pi$. For example, Wallis's product: https://en.wikipedia.org/wiki/Wallis_product . From there one must demonstrate that $\pi$ so defined satisfies $\pi=\frac{C}{D}$.
Similarly, the definition promulgated by Edwards must result in $\pi=\frac{C}{D}$. I'm sure I could provide persuasive arguments toward this end. What I would like to know is whether there is an established convention for demonstrating that Edwards's definition of $\pi$ is equivalent to the traditional form I stated above.
In particular, I am interested to know how this has been done without the use of trigonometry nor calculus.
Edit to add: This is my heuristic argument showing $2\pi r=C$. Assuming $A=\pi r^2$.
For a small change in radius $\Delta r$ there will be a corresponding change in circumference $\Delta C$. The change in area will be the area of an annulus
$\Delta A=\pi(r-\Delta r)^{2}-\pi r^{2}=\pi(2r\Delta r+\Delta r^{2})=(C+\varepsilon)\Delta r$.
Where $0<\varepsilon<\Delta C$.
The last expression follows from the observation that $(C+\Delta C)\Delta r>\Delta A>C\Delta r$. That is, $\Delta A$ falls somewhere between the area of a rectangle $C\times\Delta r$ and the area of a rectangle $(C+\Delta C)\times\Delta r$. The last two expressions in the above equivalence result in
$2\pi r+\pi\Delta r=C+\varepsilon$.
Clearly $\varepsilon\to0$ as $\Delta r\to0$. So what remains is
$2\pi r=C$.