What do we use the diameter of a circle for?

Question

Today, the 14^th of March, is $\pi$ day (in America the date is 3-14 - in the rest of the world today's date is 14-3).

We define $\pi=\frac{C}{d}$. Yet, that seems to be the last time we use the diameter $d$, immediately switching to the radius $r$, e.g. $C=2\pi r$, $A=\pi r^2$, etc. I know I've seen lots of formulae using $r$, but I'm not aware of seeing any other formulae using $d$.

What formulae (other than the identity for $\pi$) use the diameter $d$ in preference to the radius $r$ as the canonical version of the formula?

Obviously, we can change any formula that uses $r$ to use $\frac{d}{2}$ instead, but that's not the point. Where do we already use $d$ in preference to $r$?

P.S. I am aware of $\tau=\frac{C}{r}$: using it we get $C=\tau r$ instead.

Answer 1

For every formulae with $ d $, we can change it to expression of $ r $, and expression in $ r $ gives us more possibilities to integers, than expression in $ d $. If you want to express it in $d$ , just change $r$ to $d/2$ .

Answer 2

Well, in modern mathematics the diameter is well established as an important notion in general metric spaces; and the main issue here is that in general, the diameter $d$ is not twice the radius $r$ (rather, $r\le d\le 2r$). So yes, plenty of formulae are out there (in metric spaces) involving the diameter instead of the radius: for instance, it is a basic fact of graph theory that in each graph there is a cycle whose length is at most $2d+1$.