The Basel Problem and Theodorus' Spiral

I've been trying to find a classical solution to the famous Basel Problem solved by Euler. To those unfamiliar the problem is to find the sum infinite series made up of the reciprocals of square numbers (ie 1+1/4+1/9+1/16+...).

The solution I'm working on makes use of good ol' Pythagorean Theorem. My idea is to essentially start with a right triangle with leg lengths 1 and 1/2. Then, I build another right triangle from the hypotenuse. The legs of this new triangle are the hypotenuse and also the length 1/3. The process is then repeated over and over again, with new legs of 1/4, 1/5, 1/6, etc. In this way, if the process is carried out infinitely, the hypotenuse should converge to a number (the square root of sum of the series.) An easier way to understand my construction is to think about the Spiral of Theodorus but with legs of the harmonic series instead of just "1".

I think this is a cool construction, and I'm wondering if there is a path to finding this sum, either by using trig on the angles formed or some other method altogether.

So I'm asking for methods of gathering information about "Theodorus-esque" constructions. Also, if you know of a strictly geometric approach to evaluating the Basel series, please share it.