Four circles of equal radii are inscribed in a square of side 56 cm, touching each other such that each side of the square is tangent to two adjacent circles. The area of the square unoccupied by the circles is

Options

A. 672
B. 505
C. 670
D. 680

Answer

To solve the problem of finding the area of the square unoccupied by the circles, we can follow these steps:

### Step 1: Calculate the area of the square
The area of a square is given by the formula:
\[ \text{Area of square} = \text{side}^2 \]
Given the side of the square is 56 cm:
\[ \text{Area of square} = 56^2 = 3136 \text{ cm}^2 \]

**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, they are arranged in such a way that they touch each other and the sides of the square. The diameter of each circle is equal to half the side length of the square:
\[ \text{Diameter of one circle} = \frac{\text{side of square}}{2} = \frac{56}{2} = 28 \text{ cm} \]
Thus, the radius \( r \) of each circle is:
\[ r = \frac{\text{Diameter}}{2} = \frac{28}{2} = 14 \text{ cm} \]

**Hint:** The radius is half of the diameter, so make sure to divide the diameter by 2 to find the radius.

### Step 3: Calculate the area of one circle
The area of a circle is given by the formula:
\[ \text{Area of circle} = \pi r^2 \]
Using \( r = 14 \text{ cm} \):
\[ \text{Area of one circle} = \pi (14)^2 = \pi \times 196 \approx 3.14 \times 196 \approx 615.44 \text{ cm}^2 \]

**Hint:** Use \( \pi \approx 3.14 \) for calculations unless specified otherwise.

### Step 4: Calculate the total area occupied by the four circles
Since there are four circles:
\[ \text{Total area of circles} = 4 \times \text{Area of one circle} = 4 \times 615.44 \approx 2461.76 \text{ cm}^2 \]

**Hint:** When calculating the total area occupied by multiple identical shapes, multiply the area of one shape by the number of shapes.

### Step 5: Calculate the area unoccupied by the circles
To find the area of the square that is unoccupied by the circles, subtract the total area of the circles from the area of the square:
\[ \text{Area unoccupied} = \text{Area of square} - \text{Total area of circles} \]
\[ \text{Area unoccupied} = 3136 - 2461.76 \approx 674.24 \text{ cm}^2 \]

**Hint:** The unoccupied area is found by subtracting the area occupied by the circles from the total area of the square.

### Final Answer
The area of the square unoccupied by the circles is approximately \( 674.24 \text{ cm}^2 \).

**Note:** The closest option to our calculated value is 672 cm², which may be due to rounding or approximation in the calculation of π.