Question

. In the given figure, OACB is a quadrant of a
circle with centre O and radius 3.5 cm. If OD = 2 cm, find the area of the shaded
region.

[CBSE Delhi 2017] ​

Answer 1

ACB is a quadrant, subtend at 90° angle at O.

So, ø = 90° , r = 3.5cm

now,

Area of quadrant OACB

22/7 × 90/360 × ( 3.5 ) ²

22×49 / 7×16

77/8 cm²

And, Area of ΔOBD

1/2 × OB × OD

1×2×3.5 / 2

7/2 cm²

Area of the shaded region = Area of quadrant OACB − Area of ΔOBD

( 77/8 - 7/2 ) cm²

49/8 cm²

HENCE, Area of shaded region = 6.125cm²

Answer 2

Solution :

i ) Dimensions of the sector OACB :

Radius ( r ) = OB = OA = 3.5 cm

sector angle ( x ) = 90°

Area of the sector = ( x/360 ) × πr²

= ( 90/360 ) × ( 22/7 ) × 3.5²

= ( 1/4 ) × ( 22/7 ) × 3.5 × 3.5

= 9.625 cm² ----( 1 )

ii ) Dimensions of the right ∆BOD,

base ( b ) = 3.5 cm

height ( h ) = OD

h = 2 cm ( given )

Area ∆BOD = ( bh )/2

= ( 3.5 × 2 )/2

= 3.5 cm² ---------( 2 )

iii ) Area of shaded region = ( 1 ) - ( 2 )

= 9.625 - 3.5

= 6.125 cm²

******