Question

PQRS is a diameter of a circle of radius 6cm. The lengths PQ, QR and RS are equal. Semi circles are drawn on PQ and QS as diameters. Find the perimeter and area of the region so obtained.
with explanation

Answer 1

 ps = 12 cm
⇒pq = qr =rs
∴ pq =qr =rs = 1/3 x ps = 1/3 x 12 = 4 cm.
qs = 2 pq
qs = 2 x 4 = 8 cm
∴ area of shaded region = area of semicircle with ps as diameter + area of semicircle with pq as diameter – area of semicircle with qs as diameter
= 1/2 [ 3.14 x 6² + 3.14 x 2² - 3.14 x 4² ]
= 1/2 [ 3.14 x 36 + 3.14 x 4 – 3.14 x 16 ] 
= 1/2[ 3.14 ( 36 + 4 – 16)]
= 1/2 ( 3.14 x 24 ) = 1/2 x 75.36 
∴ area of shaded region = 37.68 cm²
Perimeter of shaded region=12*22/7 =264/7 

Answer 2

 PS = 12 cm
As PQ = QR =RS 
∴ PQ =QR =RS = 1/3 x PS = 1/3 x 12 = 4 cm.
QS = 2 PQ 
QS = 2 x 4 = 8 cm
∴ Area of shaded region = Area of semicircle with PS as diameter + Area of semicircle with PQ as diameter – Area of semicircle with QS as diameter.
= ½ [ 3.14 x 6 2 + 3.14 x 2 2 - 3.14 x 4 2 ]
= ½ [ 3.14 x 36 + 3.14 x 4 – 3.14 x 16 ] 
= ½ [ 3.14 ( 36 + 4 – 16)]
= ½ ( 3.14 x 24 ) = ½ x 75.36 
∴ Area of shaded region = 37.68 cm 2