Question

PQRS is a diameter of a circle of radius 6 cm. The lengths PQ, QR and RS are equal. Semicircles are drawn with PQ and QS as diameters, as shown in the given figure. If PS = 12 cm, find the perimeter and area of the shaded region.

Answer 1

Perimeter (circumference of the circle) = 2πr
We know:
Perimeter of a semicircular arc = πr
Now,
For the arc PTS, radius is 6 cm.
∴ Circumference of the semicircle PTS =πr = 6π cm

For the arc QES, radius is 4 cm.
​∴ Circumference of the semicircle QES = πr = 4π cm

For the arc PBQ, radius is 2 cm.
∴ Circumference of the semicircle PBQ = πr = 2π cm

Now,
Perimeter of the shaded region == 6π + 4π + 2π

= 12πcm

= 12 ×3.14

= 37.68 cm

Area of the semicircle PBQ `=1/2 pi"r"^2`

`= 1/2xx3..14xx2xx2`

= 6.28 cm² 

Area of the semicircle PTS `= 1/2pi"r"^2`

`=1/2xx3.14xx6xx6`

= 56.52 cm²

Area of the semicircles QES `= 1/2pi"r"^2`

`=1/2xx3.14xx4xx4`

= 25.12 cm² 

Area of the shaded region = Area of the semicircle PBQ + Area of the semicircle PTS - Area of the semicircle QES = 6.28 + 56.52 - 25.12 =  37.68 cm²