Allow me to clarify ...
With a circle of circumference $c$ and diameter $d$, $\pi$ makes an appearance as $\frac{c}{d}$ even though the equation of a circle, $(x - h)^2 + (y - k)^2 = r^2$, doesn't contain $\pi$.
I know that for an ellipse there is a constant, $C = 2a$ where the combined distance of a point $P$ on the ellipse from the two foci = twice the major radius. I was wonder though whether there's anything $\pi$-ish about an ellipse's circumference and "diameter"? Is it possible that if $R$ is the mean of the major and minor radii of an ellipse and $C$ the ellipse's circumference, $\frac{C}{2R} \approx \pi$?
Do nfr things
There is a $\pi$ in both the area and perimeter formulas.