geometry
posted by The Boss .
Square ABCD and circle T have equal areas and share the same center O.The circle intersects side AB at points E and F.Given that EF=√(1600400π),what is the radius of T ?

Hmmm. Let's set up a diagram.
Drop a perpendicular from O to AB to intersect at P.
Let OA intersect circle T at Q.
Now, let
b = AE
h = OP
a = AQ
OA is the radius r of T.
EF = 20√(4π)
let d be the diagonal of the square, s√2
Now, since ABCD and T have the same area,
s^2 = πr^2
s = r√π
c = 1/2 EF = 10√(4π)
b+c = s/2 = r/2 √π, so
b = r/2 √π  10√(4π)
a+r = d/2 = s/√2, so
a = r  r√(π/2)
since we have two secants from A,
b(sb) = a(da)
plug it all in, and you get r=20. 
I sketched the situation so that, for the square, point A is in quadrant I and B in in quad II
let the radius of the circle be r
area of circle = πr^2
then the area of square is πr^2
and each side of the square is r√π
we can call A(r√π/2 , r√π/2)
EA = r√π/2  (1/2)EF
= r√π/2  (1/2)√(1600  400π)
= r√π/2  (1/2)(20)√(4  π)
= r√π/2  10√(4π)
so now:
r^2 = (r√π/2)^2 + (r√π/2  10√(4π) )^2
arghhhh!!!!!
I then multiplied by 2, simplified and got
r^2(π2)  20π√(4√) + 800200π = 0
which is a quadratic in r, with
a=π2
b=20π√(4π)
c = 800200π
I used my calculator and the quadratic formula to get
r = 47.85 or r = 3.14285
checking:
if r = 3.14285..
area of circle = 31.0311...
side of square = r√π = 3.14285√π = 5.5705..
area of square = 31.0311... YEAHHHHH!
if r = 47.85..
area of circle = appr 7193
side of square = 47.85√π = 84.81..
area of square = appr. 7193
both answers are valid,
r = appr 47.85 or r = 3.14285 
Hmmm. I musta blown it somewhere. Of course, if r=3.14, there's no way that EF could be √(1600400π) = 18.53

"overthinking" again.

Steve, my hunch is that you are correct,
your answer came out "too nice" and clearly mine does not work.