Question

In figure, OA = 7 cm, AB = 3.5 cm. Calculate the perimeter and area of the shaded region.

Answer 1

Given:

• OA = 7 cm
• AB = 3.5 cm 
• ∠AOB = 90°

We have two quadrants (quarter circles):

Outer radius = OB

= OA + AB 

= 7 + 3.5 

= 10.5 cm

Inner radius = OA = 7 cm

We need to find:

1. Area of shaded region area between the two quadrants
2. Perimeter of shaded region

1. Area of shaded region 

Area of shaded region = Area of outer quadrant – Area of inner quadrant

`A = 1/4 π(R^2 - r^2)`

Substitute:

R = 10.5, r = 7, π = 3.14

`A = 1/4 xx 3.14 xx (10.5^2 - 7^2)`

A = 0.785 × (110.25 – 49)

= 0.785 × 61.25

= 48.08125

A = 48.1 cm²   ...(Approximately 48.125 cm²)

Area = 48.125 cm²

2. Perimeter of shaded region

Perimeter = (Outer arc) + (Inner arc) + 2 × (Difference of radii)

`P = 1/4 xx 2πR + 1/4 xx 2πr + 2(R - r)`

`P = 1/2π(R + r) + 2(R - r)`

Substitute values:

`P = 1/2 xx 3.14 xx (10.5 + 7) + 2(10.5 - 7)`

P = 1.57 × 17.5 + 2(3.5)

P = 27.475 + 7 = 34.475

P = 34.5 cm   ...(Approximately)

Perimeter = 34.5 cm