Let A, B, C, D, E, and F (in that order) lie on
a horizontal line such that AE is the diameter
of the upper semicircle with center C and BF is
the diameter of the lower semicircle with center
D. Let G be the point on the upper semicircle
such that GF is tangent to the upper semicircle.
The sides of the enclosing rectangle are parallel
and perpendicular to the line GF.

Let x = |CD| and t the measure of angle CFG. Then

  A = [2 + x cos(t)][2 - x sin(t)]             (1)

    where

              1
  sin(t) = -------                             (2)
            1 + x

Plugging x from (2) into (1) gives

       cos(t)^3 + 2 sin(t)[1 + sin(t)]
  A = ---------------------------------        (3)
                    sin(t)

Setting dA/dt = 0 gives

  2 cos(t)^3 - 2 cos(t)^2 - 3 cos(t) + 2 = 0   (4)

Of the three real roots of this equation, only
one lies in [-1,1]. Therefore,

  cos(t) ~= 0.57318274451645

  t ~= 55.02753223576 degrees

  x ~= 0.22036411444

Plugging these values into (1) or (3) gives
the ratio

  A/4 ~= 0.967166262

This value agrees with what I get using Sketchpad.