Very nice work Steven.  You beat me by 1 1/2 hours.  I choose to use the 2D analogy outlined in previous posts and reduced the problem to maximizing the area of a circle of radius r, contained (fully or partially) in an isosceles triangle of base 2a and height h.  

Given my very rusty calculus, I used a spreadsheet with parameters for a and h, which also allowed calculation of confirming factors such as chord length to check the calculations.

In the end, I found the exact same transition points (1-2, 2-3, 3-4) as Steven for his two given cases.  However, I found slightly different answers for the maximum.  2.25 vs. Steven's 2.55 and 1.626 vs. Steven's 1.73. 

Also, I explored the limits of a vs. h a bit.  for large h/a, the maximum tends to the transition 1-2 point, which makes sense. I suspect that for very small h/a, the solution tends towards the 3-4 transition, but the trend is much slower and this statement is more difficult to support as I keep ruining into resolution issues with Excel.