the radii of the two circles

Let radius of the inner circle be r, by cosine rule
A^2 = r^2 + r^2 - 2(r)(r) cos (22.5 + 22.5)
A^2 = 2r^2(1 - cos 45) = 2r^2 [1 - (sqrt 2)/2]
so r = A /sqrt { 2[1 - (sqrt 2)/2]}.
The inner triangle is an isos. triangle, so its base angle = (180 - 45)/2 = 67.5
The outer triangle is an equilateral triangle, so one of its angle = 60
So the obtuse angle = 67.5 + 60 = 127.5.
Let radius of outer circle be R,
by sine rule
R/sin 127.5 = A/sin 22.5
so R = A sin 127.5/sin 22.5