The Universal parabolic constant is "the ratio between the arc length of the parabolic segment formed by the latus rectum to the focal parameter". It can be seen as '$\pi$ but for parabola'.
The universal parabolic constant can be shown to be equal to $\ln(1+\sqrt{2})+\sqrt{2}$. This is a transcendental number, but it is the sum of an algebraic number and the log of an algebraic number. I think that almost all real numbers are not of that form.
No such simple representation is known for the "universal circle constant" $\pi$ (at least not to me), but has this already been proven?
More formally: do there exist $a,b\in \mathbb{R}$, such that $\pi = a+\ln b$, and $a$ and $b$ are both algebraic?