The area of the circle is: \(2^2.3,14=12,56\left(cm^2\right)\)

Because the shaded part has the area equals to each other and in the figure => The rectangle's area is 1/2 the area of the circle (2.1/4=1/2)

So the area of the rectangle is: \(12,56.\dfrac{1}{2}=6,28\left(cm^2\right)\)

From the figure, the width of the. rectangle = the radius of the circle => The width of the rectangle is \(2cm\). So the side AB = \(6,28:2=3,14\left(cm\right)\)

So the answer is: \(3,14cm\)

Because A and B are center of two circles respectively. Let M, N denote intersections of A, B and two circles respectively

=> AM = BN = 2cm

Again: Cover MN and NM so AN = BM = 2cm - MN

So AB = 2cm + 2cm - MN = 2cm + AN = 2cm + BM

Call the radius of center circle A is AO, B is BI.

We have AO + BI = 2 + 2 = 4 (cm).

<=> AO + BI - OI = AB = 4 - (2:2) = 4 - 1 = 3 (cm).

That's my idea :((