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Question

In the given figure, APB and CQD are semicircles of diameter 7 cm each, while ARC an BSD are semicircles of diameter 14 cm each. Find the
(i) perimeter, (ii) area of the shaded region. [CBSE 2011]

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Solution

(i) Perimeter of shaded region = Length of the arc APB + Length of the arc ARC + Length of the arc BSD + Length of the arc CQD
=12×2πr1+12×2πr2+12×2πr3+12×2πr4=12×2π72+12×2π7+12×2π7+12×2π72=2π72+2π7=2π72+7=2×227×212=66 cm

(ii) Area of shaded region = Area of the arc ARC + Area of the arc BSD − (Area of the arc APB + Area of the arc CQD)
=12×πr12+12×πr22-12×πr32+12×πr42=12×π72+12×π72-12×π722+12×π722=π72-π722=π49-494=227×1474=115.5 cm2

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