First order definition of $\pi$

$e$ has a very short first-order definition. It's the only constant that makes this true: $$\forall x,e^x\ge x+1$$ What about $\pi$? What's the shortest first-order definition we could give it?

I'm including things like addition, subtraction, multiplication, division, (real) exponentiation, roots, inequalities, etc. $\forall$ and $\exists$ range over $\Bbb R$. If you want to use differentiation or limits, expand it into its $\epsilon-\delta$ form.

The best definition of the sort you require I can think of is by Stirling's factorial approximation:

$$\forall n \geq 1, \pi \leq \frac{e^{2n}(n!)^2}{2 n^{2n+1}}$$

I don't think there is any such definition that doesn't use $e$ though.

Added an illustration (the convergence is very slow, but it doesn't matter for the OP's question):

$ \pi < 4$ and for every integer $n \ge 3$ there is a complex $n$-th root of unity $\omega_n$ such that $$n |\omega_n -1| < 2\pi < (n+1) |\omega_n -1|$$