In the given figure three circles, each of radius 3.5 cm are placed in such a way that each circle touches the other two. The area of the portion enclosed by the circles is: (Up to two decimal places)
Show answer & explanation
Correct answer: B
Explanation
The Correct answer is Option 2 i.e. 1.97 cm2.
Given that, the three circles are in such a way that each touches the other two now. We join the center of all three circles to each other by a line segment. Since the radius of each circle is 3.5 cm,
So, AB = 2 × Radius of circle
⇒ 2 × 3.5 = 7 cm
⇒ AC = BC = AB = 7 cm
This shows that ΔABC is an equilateral triangle with a side of 7 cm.
We know that each angle between two adjacent sides of an equilateral triangle is 60∘.
Area of the sector with angle ∠A = 60∘
⇒ ∠A/360∘ × πr2 = 60∘/360∘ × π × (3.5)2
So, the area of each sector = 3 × Area of the sector with angle A
⇒ 3 × 60∘/360∘ × π × (3.5)2
⇒ 1/2 × 22/7 × 3.5 × 3.5
⇒ 11 × 5/10 × 35/10 = 11/2 × 7/2
⇒ 77/4 = 19.25 cm2
Area of ΔABC = √3/4 × Side2 =√3/4 × 72 = 21.21
Area of the shaded region = Area of the equilateral triangle - Area of three sectors
⇒ 21.2176 − 1925 = 1.97 cm2