Question

The boundary of the shaded region in figure consists of three semi-circular arcs, the smaller ones being equal. If the diameter of the larger arc is 10 cm, calculate:

1. the length of the boundary,
2. the area of the shaded region (Take π = 3.14).

Answer 1

Given:

Diameter of larger semicircle = 10 cm

⇒ Radius R = 5 cm

Two smaller semicircles are equal.

π = 3.14

Since the two smaller semicircles together make up the same base as the large one:

Diameter of each smaller semicircle = `10/2` = 5 cm

⇒ Radius of each smaller semicircle = 2.5 cm

i. Length of the boundary

Length of semicircle = πr

Large semicircle:

πR = 3.14 × 5

= 15.7 cm

Two small semicircles:

Each:

3.14 × 2.5 = 7.85 cm

Two:

7.85 + 7.85 = 15.7 cm

Total boundary length:

15.7 + 15.7 = 31.4 cm

ii. Area of shaded region

Area of semicircle = `1/2 πr^2`

Area of large semicircle:

`1/2 xx 3.14 xx 2.5^2`

= `1/2 xx 3.14 xx 6.25`

= 9.8125

Two:

19.625 cm²

Since the smaller semicircles are removed from the larger one:

39.25 – 19.625 = 39.25 cm²