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

`\frac A{\pi(12\,\mathrm{cm})^2}=(45^\circ)/(360^\circ)\implies A=18\pi\,\mathrm{cm}^2`

`\frac a{\pi(8\,\mathrm{cm})^2}=(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`.