Assuming any commonplace, but rigorous, definitions and properties of $\pi$, what could be a simple, rigorous and non-numerical argument for the following inequality?
$$\pi\neq\sqrt{2}+\sqrt{3}$$
By simple I mean within the framework of a typical elementary analysis course at university, say, and thereby excluding deeper properties like transcendence etc.
By non-numerical, I mean arguments that avoid calculating approximations.
Find a shape which contains a circle of diameter $1$ and has perimeter $\sqrt{2}+\sqrt{3}$. By some isoperimetric inequalities, this shows that $\pi < \sqrt{2} + \sqrt{3}$.