Question

Calculate the area of the shaded region.

Answer 1

Let $A$ 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 $a$ be the area of the sector in the smaller circle (radius 8 cm) with the same central angle. The area you want is $A-a$.

We have

$\dfrac A{\pi(12\,\mathrm{cm})^2}=\dfrac{45^\circ}{360^\circ}\implies A=18\pi\,\mathrm{cm}^2$

$\dfrac a{\pi(8\,\mathrm{cm})^2}=\dfrac{45^\circ}{360^\circ}\implies a=8\pi\,\mathrm{cm}^2$

So the area of the shaded region is $18\pi-8\pi=10\pi$.