In the given 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.
А
B
C
Floor of a room is of dimani
Answers
Answer:
Given: ΔABC is an equilateral triangle of side 4cm.
In ΔBDO we have ,
cos∠OBD=
OB
BD
⇒cos30
∘
=
OB
2
[∵∠OBD=30
∘
]
⇒
2
3
=
OB
2
⇒OB=
3
4
∴ OP=OB+BP
⇒R=(
3
4
+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(
3
4
+2)
2
−3×π×2
2
+6×(
360
60
×π×2
2
)−
4
3
×4
2
]cm
2
=[π(
3
16
+4+
3
16
)−12π+4π−4
3
]cm
2
=[π(
3
4
+
3
16
)−4
3
]cm
2
=[
3
4π
(4
3
+1)−4
3
]cm
2