You rock, Nick!
Ok, here's what I got:
Assuming a 14" opening, I get an overall hypotenuse length, x, of 10.82". Using tangent, I get an angle, Ө, of 40.3°. Given an existing opening, minus 1/4" all around for the new back to sit in, of 7-3/4" (half of which is 3-7/8"), using sine I get a short hypotenuse, y, of 5.99", meaning the angled extension, z, is going to be 4.83" wide, and the overall depth is going to be 3.68".
Now, I've gotta do a reality check and make sure the assumed dimensions are good, but that looks about right, doesn't it?
Thanks nick!