Question

Four equal circles, each of radius 7 cm, touch each other and a square ABCD is formed through the centres, A, B, C, D of these circles as shown in the figure. Then the area of the shaded region is

• A. 119 sq. cm
• B. 42 sq. cm
• C. 157.5 sq. cm
• D. None of these

Answer 1

The Correct Option is B

Step 1: Determine the side length of the square.

Since the circles touch each other, the distance between the centers of two adjacent circles is equal to twice the radius of the circles. Thus, the side length of the square is:

\[ \text{Side length of square} = 2 \cdot 7 = 14 \, \text{cm}. \]

Step 2: Calculate the area of the square.

The area of a square is given by:

\[ \text{Area of square} = \text{side}^2 = 14^2 = 196 \, \text{sq. cm}. \]

Step 3: Calculate the total area of the four quadrants of the circles.

Each circle has a radius of \( 7 \, \text{cm} \), so the area of one circle is:

\[ \text{Area of one circle} = \pi r^2 = \pi (7)^2 = \pi \cdot 49. \]

Using \( \pi = \frac{22}{7} \):

\[ \text{Area of one circle} = \frac{22}{7} \cdot 49 = 22 \cdot 7 = 154 \, \text{sq. cm}. \]

Each quadrant is one-fourth of a circle, and there are four quadrants (one from each circle). Thus, the total area of the four quadrants is:

\[ \text{Total area of quadrants} = 4 \cdot \frac{1}{4} \cdot 154 = 154 \, \text{sq. cm}. \]

Step 4: Find the area of the shaded region.

The shaded region is the area of the square minus the total area of the four quadrants:

\[ \text{Shaded area} = \text{Area of square} - \text{Total area of quadrants}. \]

Substitute the values:

\[ \text{Shaded area} = 196 - 154 = 42 \, \text{sq. cm}. \]

Final Answer: The area of the shaded region is \( \mathbf{42 \, \text{sq. cm}} \), which corresponds to option \( \mathbf{(2)} \).