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 πand√3
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Solution
Given: ΔABC is an equilateral triangle of side 4cm.
In ΔBDO we have ,
cos∠OBD=BDOB
⇒cos30∘=2OB[∵∠OBD=30∘]
⇒√32=2OB
⇒OB=4√3
Produce OB such that it meets the larger circle at P. OP is the radius of larger circle. ∴OP=OB+BP
⇒R=(4√3+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