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. [ Take $\pi$ = 3.14]

Answer 1

Observing the figure.

We have that:
PS = 12 CM
as
PQ = QR = RS=$\frac{1}{3}$×PS=$\frac{1}{3}$×12=4 cm
and
QS=2PQ
QS=2×4 = 8 cm

we calculate the area of shaded region:

A= area of a semicircle with PS as diameter + area of a semicircle with PQ as diameter – the area of a semicircle with QS as diameter;

= $\frac{1}{2}$ [ 3.14 x 6² + 3.14 × 2² - 3.14 × 4² ]
= $\frac{1}{2}$ [ 3.14 ×36 + 3.14 ×4 – 3.14 ×16 ]
= $\frac{1}{2}$[ 3.14 ( 36 + 4 – 16)]
= $\frac{1}{2}$ ( 3.14 × 24 )
= $\frac{1}{2}$ × 75.36
= 37.68 cm²

The area of shaded region = 37.68 cm².

The perimeter of the shaded region = Arc of the semicircle of radius 6 +Arc of the semicircle of radius 4 + Arc of the semicircle of radius 2

= (6π+4π+2π) = 12π

= 12×$\frac{22}{7}$ = $\frac{264}{7}$ = 37.71 cm