In the figure (ii) given below, the boundary of the shaded region in the given diagram consists of three semicircular arcs, the smaller being equal. If the diameter of the larger one is 10 cm, calculate.

(i) the length of the boundary.

(ii) the area of the shaded region. (Take π to be 3.14)

From figure,

Diameter of big semi-circle = 10 cm.

Radius of big semi-circle (R) = $\dfrac{10}{2}$ = 5 cm,

Diameter of small semi-circle = 5 cm.

Radius of each smaller semi-circle (r) = $\dfrac{5}{2}$ = 2.5 cm.

(i) Length of boundary = Circumference of bigger semi-circle + 2 x circumference of smaller semi-circles

= πR + πr + πr

= π(R + 2r)

= 3.14(5 + 2 × $\dfrac{5}{2}$)

= 3.14(5 + 5)

= 3.14 × 10

= 31.4 cm.

Hence, length of boundary = 31.4 cm.

(ii) From figure,

Area of shaded region = Area of bigger semi-circle + Area of one smaller semi-circle – Area of other smaller semi-circle

$= \dfrac{πR^2}{2} + \dfrac{πr^2}{2} - \dfrac{πr^2}{2} \\[1em] = \dfrac{πR^2}{2} \\[1em] = \dfrac{3.14 \times 5^2}{2} \\[1em] = \dfrac{3.14 \times 25}{2} \\[1em] = 1.57 \times 25 \\[1em] = 39.25 \text{ cm}^2.$

Hence, area of shaded region = 39.25 cm².