Let [tex]A[/tex] be the area of the sector in the larger circle (radius 12 cm) whose central angle is subtended by the labeled arc with measure 45 deg, and let [tex]a[/tex] be the area of the sector in the smaller circle (radius 8 cm) with the same central angle. The area you want is [tex]A-a[/tex].
We have
[tex]\dfrac A{\pi(12\,\mathrm{cm})^2}=\dfrac{45^\circ}{360^\circ}\implies A=18\pi\,\mathrm{cm}^2[/tex]
[tex]\dfrac a{\pi(8\,\mathrm{cm})^2}=\dfrac{45^\circ}{360^\circ}\implies a=8\pi\,\mathrm{cm}^2[/tex]
So the area of the shaded region is [tex]18\pi-8\pi=10\pi[/tex].
