Okay I was wrong. I can’t delete the post but I’ve accepted it’s a dumb question and moved on :)
Just to add my own exhortations to the above.
Mathematical constants are fixed, and are true in all possible worlds, unlike physical constants like Planck's. It seems like you're asking for a physical explanation, which is no better than asking the same for deductive logic.
Philosophers can ask questions about mathematics. e.g. the metaphysics or epistemology of mathematics. Even, why the golden ratio is aesthetically pleasing.
Not all questions have answers, beyond just because.
I wonder why you asked this question, and why 4/5=0.8 does not puzzle you.
In answer to the question as it is, I think by"arbitrary" you could mean unmemorable. Even-though you can derive it quite easily via e.g. geometry.
It sort of sounds like you're getting stuck on the numerical expression, which is subjective because Pi is also 11.00100100001111110110... and 3.243F6A8885A308D313198A2E0... [See π in Different Bases]
This is a little bit outside of my field, but I'm going to go out on a limb and say Pi is not arbitrary, but is the ratio of the circumference of a circle to it's diameter, always.
When I googled "arbitrary" in relation to Pi, I came up with many results, but none suggesting that Pi itself is arbitrary. Possibly I am misunderstanding arbitrary in the context the context you are using it...
despite pi's definition of being C/D of a circle on the Cartesian plane, its value working out to be 3.1415... seems to be arbitrary.
There is nothing arbitrary about the value of pi working out to be 3.1415... . Mathematicians do not meet up to fix the values of constants!
it seems arbitrary that C/D turns out to be 3.1415... rather than, say, 4, or any other number.
It could not be anything else than 3.1415... .
Is there a better way to describe this observation?
Your "observation" seems to be a misconception.
Other remark. Even though your question doesn't admit a very profound answer, at least you are questioning yourself. What are numbers? Why can we use numbers to represent the ratio of two planar "lengths"? These are important questions that have non-trivial answers.
Flat space is only one among a number of different kinds of spaces.
We live on a sphere, so, in fact, none of the circles we actually see on the globe have pi as the ratio of their circumference to their diameter. Circles on a sphere do not have this as a fixed ratio.
But we have the mental image of a flat space, where pi is well defined by this relation. How arbitrary was that choice?
If it is to some degree arbitrary, why should any particular part of its structure be less so? For instance, the length of a diagonal in flat space is just as irrational and no more regular than pi.