The given figure shows sector OAB with centre O and radius 54cm. Another circle XYZ with centre P, is enclosed by the sector OAB.the angle AOB=60 degree . Find the area of OXPY.

Solⁿ: Given: sector OAB with centre O and radius 54cm.

                      <AOB=60⁰

                        OZ=OB=OA=54cm

Construction: join O and P.

Let XP=r, then OP=54-r

In ΔOXP  right angled at X,

        <XOP=30⁰

          Sin30⁰=XP/OP=r/54-r=1/2     then r=18cm-----------(1)

          OP=54-18=36cm

          So,XP²+OX²=OP²

          18²+OX²=36² => OX=18√3cm---------------(2)

Area of OXPY quadrilateral=2xarea of ΔOXP

                                              =2x ½ x 18√3 x 18 = 561.2cm(ans.)

Comment: The radius of a circle drawn inside a sector is 1/3 of the mother circle