Let θ be the central angle whose endpoints are on the shorter side of the rectangle
The area of the sector formed by this angle = ( r^2/2 ) ( θ)
And the area of the isosceles triangle formed by the two radii and the shorter side of the rectangle =
(r^2/2) sin ( θ)
So....the area between the shorter side of the rectangle and the circumference is given by
4 = (r/2/2) ( θ - sin θ)
4 = (5^2/2) ( θ -sin θ)
4 = (25/2) ( θ - sin θ)
θ - sin θ = 8/25
Using a little technology to solve this for θ ≈ 1.27721 rads ≈ 73.18°
From the Law of Cosines we can find the shorter side of the rectangle as
√ [ 5^2 + 5^2 - 2(5)(5)cos(73.18°) ] = √[50 -50cos(73.18°)] ≈ 5.96 cm
And using the Pythagorean Theorem, we can find the longer side of the rectangle as
√[10^2 - 5.96^2 ] ≈ 8.03 cm
So.....the area of the rectangle will be the product of these two sides ≈ 5.96 * 8.03 ≈ 47.86 cm^2
Here's a pic :
