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Question

In the figure, three circles of radius 2 cm touch one another externally .These circles are circumscribed by a circle of radius R cm . Find the value of R and the area of the shaded region in the term of πand3
1010012_10d07d6a59324e95b77b1cf91121f681.png

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Solution

Given: ΔABC is an equilateral triangle of side 4cm.

In ΔBDO we have ,

cosOBD=BDOB

cos30=2OB[OBD=30]

32=2OB

OB=43

Produce OB such that it meets the larger circle at P. OP is the radius of larger circle.
OP=OB+BP

R=(43+2)cm

Area of the shaded region = Area of the larger circle of radius R - 3 × Area of smaller circle of radius 2 cm + 3(Area of a sector of angle 60 in a circle of radius 2 cm) - [Area of ΔABC - 3(Area of sector of angle 60 in a circle of radius 2cm)]

Area of the shaded region = Area of the larger circle of radius R - 3 × Area of smaller circle of radius 2 cm + 6 × Area of a sector angle 60 in a circle of radius of 2 - Area of ΔABC

=π(43+2)23×π×22+6×(60360×π×22)34×42cm2

=[π(163+4+163)12π+4π43]cm2

=[π(43+163)43]cm2

=[4π3(43+1)43]cm2

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