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In figure arcs have been drawn with radi...

In figure arcs have been drawn with radii 14 cm each and with centres P, Q and R. Find the area of the shaded region .

Text Solution

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Given that radii of each arc ( r) = 14cm
Now, area of the sector with central `angleP = (angleP)/(360^(@))xxpir^(2)`
= ` (angleP)/(360^(@))xxpixx(14)^(2)cm^(2)`
[`:.` area of any sector with central angle `theta` and radius r = `(pir^(2))/(360^(@))xxtheta`]
Area of the sector with central central angle = `(angleQ)/(360^(@))xxpir^(2)= (angleQ)/(360^(@))xxpixx(14)^(2) cm^(2)`
and area of the sector with central angle R = `(angleR)/(360^(@))xxpir^(2)=(angleR)/(360^(@))xxpixx(14)^(2) cm^(2)`
Therefore, sum of the areas (in `cm^(2)` ) of three sectors
= `(angleP)/(360^(@))xxpixx(14)^(2) + (angletheta)/(360^(@))xxpixx(14)^(2)+(angleR)/(360^(@))xxpixx(14)^(2)`
= `(angleP+angleQ + angleR)/(360)xx196xxpi= (180^(@))/(360^(@))xx196pi cm^(2)` [since, sum of all interior angles in any triangle is `180^(@)`]
= `98 pi cm^(2) = 98 xx (22)/(7)`
= `14xx22 = 308 cm^(2)`
Hence, the required area of the shaded region is `308 cm^(2)` .
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