Radius of a card disc from an area ratio after a quarter sector is cut away

A thin circular card has centre \(O\) and radius \(R\text{ cm}\), where \(R>4\). A circle of radius \(4\text{ cm}\), also centred at \(O\), is drawn on the card and the region inside it is shaded, as shown in the diagram below.

Two radii, \(OA\) and \(OB\), of the card are drawn so that \(A\hat{O}B=\frac{\pi}{2}\). The card is cut along \(OA\) and \(OB\) and the quarter sector \(AOB\) is discarded, leaving the shape shown below.

For the card that remains, the area of the unshaded part is one and a half times the area of the shaded part. Find the exact value of \(R\) in the form \(\sqrt{k}\), where \(k\) is an integer.[4]