Question

In the following figure, A is the centre of the arc of the circle. Find the perimeter and the area of the shaded region, where length and breadth are 12 cm and 7 cm respectively.

Answer 1

1. Perimeter of the shaded region

Step 1: Identify the components of the perimeter

The perimeter of the shaded region consists of:

1. The bottom side of the rectangle: 12 cm.

2. The two vertical sides of the rectangle: 7 cm + 7 cm = 14 cm.

3. The two parts of the top side of the rectangle: Since the semicircle's diameter is 7 cm and it is centered on the 12 cm side, the remaining straight top sections are 12 – 7 = 5 cm.

4. The length of the semicircular arc: Arc length = π × r.

Step 2: Calculate the arc length

Using `π = 22/7`:

Arc length = `22/7 xx 3.5`

= `22/7 xx 7/2`

= 11 cm

Step 3: Sum all components

Total perimeter = 12 + 7 + 7 + 5 + 11

Total perimeter = 42 cm

The perimeter of the shaded region is 42 cm.

2. Area of the shaded region

Step 1: Calculate the area of the rectangle

Area of rectangle = Length × Breadth

Area of rectangle = 12 × 7 = 84 cm²

Step 2: Calculate the area of the semicircle

Area of semicircle = `1/2 xx π xx r^2`

Area of semicircle = `1/2 xx 22/7 xx (3.5)^2`

Area of semicircle = `1/2 xx 22/7 xx 49/4`

= `(11 xx 7)/4`

= 19.25 cm²

Step 3: Subtract the semicircle area from the rectangle area

Area of shaded region = Area of rectangle – Area of semicircle

Area of shaded region = 84 – 19.25 = 64.75 cm²

The area of the shaded region is 64.75 cm².