IB AA HL May 2026 P1 TZA Q6

Consider a piece of paper in the shape of a circle with centre \(O\) and radius \(r\text{ cm}\), where \(r>6\). A second circle with centre \(O\) and radius \(6\text{ cm}\) is drawn inside the first and shaded, as shown in the following diagram.

The points \(A\) and \(B\) are on the circumference of the larger circle such that the acute angle \(A\hat{O}B=\frac{\pi}{3}\). The paper is cut along the lines \(AO\) and \(BO\) and the sector \(AOB\) with a central angle of \(\frac{\pi}{3}\) is removed as shown in the following diagram.

After removing the sector, it is now given that \(\frac{\text{the area of the unshaded region}}{\text{the area of the shaded region}}=\frac23\). Find the value of \(r\), giving your answer in the form \(\sqrt{k}\), where \(k\in\mathbb{Z}\).[4]