Answer: \(\theta=\dfrac{\pi}{7}\); perimeter \(=4+5\dfrac{\pi}{7}+\sqrt{106-90\cos(\pi/7)}\approx11.2\text{ cm}\).

We start with the main method. Use the measurements and relationships shown in the diagram, then translate them into algebraic or trigonometric equations. Use the right-triangle information to find the angle in radians, then apply arc length and sector area formulae.

The outer radius is \(OB=OA+AB=5+4=9\) cm. The inner radius is \(OA=5\) cm.

The shaded region \(ABCD\) is the difference between two sectors with the same angle \(\theta\). Therefore

\(\frac12(9^2)\theta-\frac12(5^2)\theta=4\pi.\)

So

\(\frac12(81-25)\theta=4\pi,\)

which gives

\(28\theta=4\pi.\)

Hence

\(\theta=\frac{\pi}{7}.\)

For the new shaded region \(ACD\), the perimeter is made from \(DC\), arc \(AD\), and \(AC\).

Since \(OC=9\) and \(OD=5\),

\(DC=4.\)

The arc \(AD\) has length

\(5\theta=5\cdot\frac{\pi}{7}=\frac{5\pi}{7}.\)

To find \(AC\), use the cosine rule in triangle \(AOC\):

\(AC^2=5^2+9^2-2(5)(9)\cos\frac{\pi}{7}.\)

Thus

\(AC=\sqrt{106-90\cos\frac{\pi}{7}}\approx4.991.\)

Therefore the perimeter is

\(4+\frac{5\pi}{7}+\sqrt{106-90\cos\frac{\pi}{7}}\approx11.2\text{ cm}.\)

This completes the solution and gives the required result.