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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. - Mathematics

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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.

In the figure alongside, OACB is a quadrant of a circle. The radius OA = 3.5 cm and OD  = 2 cm. Calculate the area of the shaded portion (Take Π = 22/7)

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Solution 1

Since OACB is a quadrant, it will subtend 90° angle at O.

Area of quadrant OACB

`= 90^@/360^@ xx pir^2`

`=1/4 xx 22/7 xx (3.5)^2 = 1/4 xx 22/7 xx(7/2)^2`

`= (11xx7xx7)/(2xx 7xx 2xx2) = 77/8 cm^2`

Area of ΔOBD

`= 1/2 xx OB xx OD`

`= 1/2 xx 3.5 xx 2`

`= 1/2 xx 7/2 xx 2`

`= 7/2 "cm"^2`

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

`= 77/8 - 7/2`

`= (77 - 28)/8`

= `49/8 "cm"^2`

Solution 2

Area of the quadrant OACB = `1/4 xx pir^2`

`= 1/4 xx 22/7 xx 3.5 xx 3.5`

`= 9.625 cm^2`

Area of the triangle OAD = `1/2 xx base xx height = 1/2 cc 3.5 xx 2 = 3.5 cm^2`

Shaded Area = Area of quadrant OACB – area of triangle OAD

` = 9.625 - 3.5 cm^2`

`= 6.125 cm^2`

Concept: Areas of Sector and Segment of a Circle
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