The area of a circle inscirbed in a square of area `2 m^(2)` is

Options

A. `(pi)/(4)m^(2)`
B. `(pi)/(2)m^(2)`
C. `pi m^(2)`
D. `2pi m^(2)`

Answer

To find the area of a circle inscribed in a square with an area of \(2 \, m^2\), we can follow these steps:

### Step 1: Find the side length of the square
The area of a square is given by the formula:
\[ \text{Area} = \text{side}^2 \]
Given that the area of the square is \(2 \, m^2\), we can find the side length (\(s\)) of the square:
\[ s^2 = 2 \implies s = \sqrt{2} \, m \]

### Step 2: Find the radius of the inscribed circle
The radius (\(r\)) of a circle inscribed in a square is half the length of the side of the square:
\[ r = \frac{s}{2} = \frac{\sqrt{2}}{2} \, m \]

### Step 3: Calculate the area of the inscribed circle
The area (\(A\)) of a circle is given by the formula:
\[ A = \pi r^2 \]
Substituting the radius we found:
\[ A = \pi \left(\frac{\sqrt{2}}{2}\right)^2 = \pi \left(\frac{2}{4}\right) = \pi \left(\frac{1}{2}\right) \, m^2 \]
Thus, the area of the inscribed circle is:
\[ A = \frac{\pi}{2} \, m^2 \]

### Final Answer
The area of the circle inscribed in a square of area \(2 \, m^2\) is \(\frac{\pi}{2} \, m^2\).

---