Four circles of equal radius are drawn with centers, A, B, C and D such that ABCD is a square of side 14 cm and the circles touch externally as in the figure. The area of the shaded region bounded by the 4 circles is: (Take $π =\frac{22}{7}$).
Answer & explanation
Correct answer: option 2
The correct answer is Option (2) → $42\, cm^2$
Given:
- ABCD is a square of side 14 cm
- Four equal circles are drawn with centers at A, B, C, D such that each touches the adjacent circles externally
- π = 22/7
- Find the shaded area in the middle.
Step 1: Radius of the circles
Since ABCD is a square of side 14 cm, and the circles touch externally, the distance between centers of adjacent circles = side of the square = 14 cm.
$\text{Radius of each circle} = \frac{\text{side of square}}{2} = \frac{14}{2} = 7 \text{ cm}$
Step 2: Area of the square
$\text{Area of square} = 14 \times 14 = 196 \text{cm}^2$
Step 3: Area of the four quarter circles inside the square
- Each corner has a quarter circle (since the center is at the corner)
- Total area of 4 quarter circles = area of 1 full circle
$\text{Area of 1 circle} = \pi r^2 = \frac{22}{7} \times 7^2 = \frac{22}{7} \times 49 = 154 \text{cm}^2$
Step 4: Area of the shaded region
- The shaded area = area of square − area covered by the 4 quarter circles
$\text{Shaded area} = 196 - 154 = 42 \text{cm}^2$