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 = 𝟐𝟐/𝟕 )

The correct option is 42 cm².

Step-by-step Explanation:

Based on the provided image, we can identify a square labeled ABCD, where the vertices A, B, C, and D serve as the centers of four identical circles that touch each other externally. The shaded region is the central area enclosed between these four circles inside the square.

1. Find the radius of the circles:
The side of the square ABCD is given as:

$$s=14\text{ cm}$$
Since the circles touch each other externally, the distance between any two adjacent centers (for example, A and B) is equal to the sum of their radii. Because all four circles are of equal radius
$r$
, the side of the square is equal to twice the radius:

$$2r=14\text{ cm}$$
Solving for the radius
$r$
:

$$r=\frac{14}{2}=7\text{ cm}$$

2. Calculate the area of the square ABCD:
The area of the square is given by the formula:

$${\text{Area}}_{\text{square}}=s^2$$
Substituting the side length of the square:

$${\text{Area}}_{\text{square}}={14}^2=196{\text{ cm}}^2$$

3. Calculate the area of the four quadrants inside the square:
Since the interior angle of a square at each vertex (A, B, C, and D) is
${90}^{\circ }$
, the portion of each circle that lies inside the square is a quadrant (one-fourth of a circle).
The area of these 4 quadrants combined is equal to the area of one full circle:

$${\text{Area}}_{\text{4 quadrants}}=4\times \left(\frac{1}{4}\pi r^2\right)=\pi r^2$$
Taking
$\pi =\frac{22}{7}$
and substituting
$r=7\text{ cm}$
:

$${\text{Area}}_{\text{4 quadrants}}=\frac{22}{7}\times 7^2=22\times 7=154{\text{ cm}}^2$$

4. Calculate the area of the shaded region:
The area of the shaded region is the difference between the total area of the square and the combined area of the 4 quadrants:

$${\text{Area}}_{\text{shaded}}={\text{Area}}_{\text{square}}-{\text{Area}}_{\text{4 quadrants}}$$
Substituting the calculated values:

$${\text{Area}}_{\text{shaded}}=196-154=42{\text{ cm}}^2$$
Thus, the area of the shaded region bounded by the 4 circles is 42 cm².