Is $\alpha = \pi^2$ algebraic over $\mathbb{Q}(\pi)$?

Question

I am working on a problem that asks if $\alpha = \pi^2$ is algebraic over $\mathbb{Q}(\pi)$ (that is "$\mathbb{Q}$ adjoin $\pi$"). I did an identical problem right before that asks for the same thing but just for the field $\mathbb{Q}$, where I concluded that $\pi^2$ is transcendental over $\mathbb{Q}$. I am leaning towards that $\pi^2$ is algebraic over $\mathbb{Q}(\pi)$ simply because $\pi \in \mathbb{Q}(\pi)$, but I am not sure that should be the right reason even if it may be true.