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In figure, arcs have been drawn of radiu...

In figure, arcs have been drawn of radius `21` `cm` each with vertices `A`,`B`,`C` and `D` of quadrilateral `ABCD` as centres . Find the area of the shaded region.

Text Solution

Verified by Experts

Given that, radius of each arc ( r) = 21 cm
Arae of sector with `angleA = (angleA)/(360^(@))xxpir^(2) = (angleA)/(360^(@))xxpixx(21)^(2) cm^(2)` [`because` area of any sector with central angle `theta` and radius r = `(pir^(2))/(360^(@))xxtheta`]
Area of sector with `angleB = (angleB)/(360^(@))xxpir^(2) = (angleB)/(360^(@)) xxpixx(21)^(2) cm^(2)`
Area of sector with `angleC= (angleC)/(360^(@))xxpir^(2) = (angleC)/(360^(@))xxpi xx(21)^(2) cm^(2)`
and area of sector with `angleD = (angleD)/(360^(@))xxpir^(2) = (angleD)/(360^(@))xxpixx(21)^(2) cm^(2)`
Therefore , sum of the areas (in `cm^(2)`) of the four sectors
= `(angleA)/(360^(@))xxpixx(21)^(2) + (angleB)/(360^(@))+ pixx(21)^(2) + (angleC)/(360^(@))xxpixx(21)^(2) + (angleD)/(360^(@))xxpi xx(21)^(2)`
= `((angleA+angleB+angleC+angleD))/(360^(@))xxpixx(21)^(2)` [`:.` sum of all interior angle in any quadrilaterl = `360^(@)`]
= ` 22xx3xx21 = 1386 cm^(2)`
Hence, required area of the shaded region is `1386 cm^(2)` .
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