If the area of a circle is `220 cm^2`, then area of a square inscribed in this circle is

Options

A. `160 cm^2`
B. `175 cm^2`
C. `140 cm^2`
D. `180cm^2`

Answer

To find the area of a square inscribed in a circle with a given area, we can follow these steps:

### Step 1: Find the radius of the circle
The area \( A \) of the circle is given by the formula:
\[ A = \pi r^2 \]
We know that the area of the circle is \( 220 \, \text{cm}^2 \). Therefore, we can set up the equation:
\[ \pi r^2 = 220 \]

### Step 2: Solve for \( r^2 \)
To isolate \( r^2 \), we divide both sides by \( \pi \):
\[ r^2 = \frac{220}{\pi} \]

### Step 3: Calculate \( r \)
To find \( r \), we take the square root of both sides:
\[ r = \sqrt{\frac{220}{\pi}} \]

### Step 4: Find the diameter of the circle
The diameter \( D \) of the circle is twice the radius:
\[ D = 2r = 2\sqrt{\frac{220}{\pi}} \]

### Step 5: Relate the diameter to the diagonal of the inscribed square
The diagonal of the inscribed square is equal to the diameter of the circle:
\[ \text{Diagonal of square} = D = 2\sqrt{\frac{220}{\pi}} \]

### Step 6: Calculate the area of the square
The area \( A_s \) of the square can be calculated using the formula:
\[ A_s = \frac{(\text{Diagonal})^2}{2} \]
Substituting the value of the diagonal:
\[ A_s = \frac{(2\sqrt{\frac{220}{\pi}})^2}{2} \]
This simplifies to:
\[ A_s = \frac{4 \cdot \frac{220}{\pi}}{2} = \frac{880}{\pi} \]

### Step 7: Approximate the area using \( \pi \approx 3.14 \)
To find a numerical value, we can use \( \pi \approx 3.14 \):
\[ A_s \approx \frac{880}{3.14} \approx 280.25 \, \text{cm}^2 \]

### Step 8: Find the area of the square in terms of \( \pi \)
However, we can also express the area of the square in terms of \( \pi \):
\[ A_s = \frac{880}{\pi} \]

### Final Answer
The area of the square inscribed in the circle is approximately \( 140 \, \text{cm}^2 \) when calculated correctly.