Question

In the following figure, the boundary of the shaded region in the given diagram consists of four semi-circular arcs, the smallest two being equal. If the diameter of the largest is 14 cm and of the smallest is 3.5 cm, calculate:

1. the length of the boundary,
2. the area of the shaded region `("Take"  π = 22/7)`.

Answer 1

Given:

Diameter of largest semicircle = 14 cm

⇒ Radius R = 7 cm

Diameter of each smallest semicircle = 3.5 cm

⇒ Radius r = 1.75 cm

`π = 22/7`

From the figure, the boundary consists of 4 semicircular arcs:

1. One large semicircle (diameter 14 cm)
2. Two small equal semicircles (diameter 3.5 cm each)
3. One middle semicircle

Step 1: Find diameter of the middle semicircle

The total base equals the diameter of the largest semicircle:

14 = 3.5 + Middle diamater + 3.5

Middle diameter = 14 – 7 = 7 cm

So, middle semicircle radius = 3.5 cm

i. Length of the boundary

Length of a semicircle = πr

Large semicircle:

`πR = 22/7 xx 7`

= 22 cm

Middle semicircle:

`π xx 3.5 = 22/7 xx 3.5`

= 11 cm

Two small semicircles:

Each:

`π xx 1.75 = 22/7 xx 1.75`

= 5.5 cm

So two:

5.5 + 5.5 = 11 cm

Total boundary length:

22 + 11 + 11 = 44 cm

ii. Area of shaded region

Area of a semicircle = `1/2 πr^2`

Area of large semicircle:

`1/2 xx 22/7 xx 7^2`

= `1/2 xx 22/7 xx 49`

= 77 cm²

Area of middle semicircle:

`1/2 xx 22/7 xx (3.5)^2`

= `1/2 xx 22/7 xx 12.25`

= 19.25 cm²

Area of two small semicircles:

Each:

`1/2 xx 22/7 xx (1.75)^2`

= 4.8125 cm²

Two:

9.625 m²

Total shaded area:

77 + 19.25 – 9.625

= 86.625 cm²