I could find many beautiful and rigorous proofs for Euler's solution to the Basel problem here Different methods to compute $\sum\limits_{k=1}^\infty \frac{1}{k^2}$ (Basel problem)
But I am curious to know whether there are proofs by using geometry.
If anyone has proofs by geometry, please do share it with us.
Funny you should ask this today. A great video video by the YouTuber 3Blue1Brown was just posted today. (Aside: I recommend all his videos.)
The proof is based on the result mentioned by "3 revs" in the MO thread mentioned by user296602 above.