Find the area of the shaded region in fig., where ABCD is a square of side 14 cm and four circles are of same radii each.

Options

A. `128 cm^(2) `
B. `192 cm^(2) `
C. `148 cm^(2) `
D. `168 cm^(2)`

Answer

To find the area of the shaded region in the square ABCD with side length 14 cm and four circles of equal radius, we can follow these steps:

### Step 1: Calculate the area of the square
The area \( A \) of a square is given by the formula:
\[ A = \text{side}^2 \]
Here, the side of the square is 14 cm. Thus,
\[ A = 14^2 = 196 \text{ cm}^2 \]

**Hint:** Remember that the area of a square is calculated by squaring the length of one of its sides.

### Step 2: Determine the radius of the circles
Since there are four circles inscribed in the square, we can infer that the circles are arranged such that their diameters fit within the side of the square. If we denote the radius of each circle as \( r \), then the total diameter of the two circles along one side of the square is equal to the side length of the square:
\[ 2r + 2r = 14 \implies 4r = 14 \implies r = \frac{14}{4} = 3.5 \text{ cm} \]

**Hint:** The diameter of a circle is twice the radius, so when fitting multiple circles along a line, consider their combined diameters.

### Step 3: Calculate the area of one circle
The area \( A_c \) of a circle is given by the formula:
\[ A_c = \pi r^2 \]
Using \( r = 3.5 \) cm, we have:
\[ A_c = \pi (3.5)^2 = \pi \times 12.25 \approx 38.48 \text{ cm}^2 \]

**Hint:** Use \( \pi \approx \frac{22}{7} \) for easier calculations if needed.

### Step 4: Calculate the total area of the four circles
Since there are four circles, the total area \( A_{total\_circles} \) is:
\[ A_{total\_circles} = 4 \times A_c = 4 \times 38.48 \approx 153.92 \text{ cm}^2 \]

**Hint:** To find the total area of multiple identical shapes, multiply the area of one shape by the number of shapes.

### Step 5: Calculate the area of the shaded region
The area of the shaded region \( A_{shaded} \) is the area of the square minus the total area of the circles:
\[ A_{shaded} = A_{square} - A_{total\_circles} = 196 - 153.92 \approx 42.08 \text{ cm}^2 \]

**Hint:** The shaded area is found by subtracting the area of the circles from the area of the square.

### Final Answer
The area of the shaded region is approximately:
\[ \boxed{42 \text{ cm}^2} \]