The center C of the larger circle lies on another circle whose diameter coincides with the chord AB of the larger circle. Find the area of the shaded region if the length AB measures 10cm.
Since AB, of length 10cm, is a diameter, the radius of the smaller circle is 5 cm. That tells you that the segment from O to P, the center of the smaller circle, is 5 cm. Now, Look at the right triangle formed by OAP. It has legs of 5 and 5 so you can find the length of OA, a radius of the larger circle.
Since $\displaystyle \angle AOB$ is inscribed in a semicircle: $\displaystyle \angle AOB = 90^o.$
As HallsofIvy pointed out: .$\displaystyle PA = PO = PB = 5$
You can now find the area of sector $\displaystyle AOB$ of the large circle
and the area of right triangle $\displaystyle AOB$,
then determine the area of the unshaded segment.