real analysis - Prove any continuous function on a 3-dim ellipsoid can be approximated by a polynomial

I'm familiar with the Weierstrass approximation theorem and some aspects of the Stone-Weierstrass theorem but I mainly only get it for closed intervals [a, b]. I am familiar with the proof that begins with showing f is continuous on $[0, 1]$ and going from there. I have a 3-d set which forms an ellipsoid and I'd like to show that any continuous function on that set can also be approximated by a polynomial. Is there a way to extend the proof of Weierstrass approximation theorem or is this way over my head.

For example, [this post] (Showing a continuous functions on a compact subset of $\mathbb{R}^3$ can be uniformly approximated by polynomials) has an example set but I can't really follow the answer. I only know introductory real analysis. I'm guessing there's a way to go from the $[0, 1]$ case to the unit cube case but I'm missing that leap.