If a square is inscribed in a circle whose area is 314 sq. cm, then the length of each side of the square is

Options

A. `5 sqrt2` cm
B. `20 sqrt2` cm
C. 10 cm
D. `10 sqrt2` cm

Answer

To find the length of each side of a square inscribed in a circle with an area of 314 sq. cm, we can follow these steps:

### Step-by-Step Solution:

1. **Understand the relationship between the circle and the square**:
- A square inscribed in a circle means that all four corners of the square touch the circle. The circle is called the circumcircle of the square.

2. **Use the area of the circle to find the radius**:
- The area \( A \) of the circle is given by the formula:
\[ A = \pi r^2 \]
- We know the area is 314 sq. cm, so we can set up the equation:
\[ \pi r^2 = 314 \]

3. **Solve for \( r^2 \)**:
- Rearranging the equation gives:
\[ r^2 = \frac{314}{\pi} \]
- Using \( \pi \approx 3.14 \):
\[ r^2 = \frac{314}{3.14} \approx 100 \]

4. **Calculate the radius \( r \)**:
- Taking the square root of both sides gives:
\[ r = \sqrt{100} = 10 \text{ cm} \]

5. **Relate the radius to the side of the square**:
- The diagonal \( d \) of the square is equal to the diameter of the circle. Since the diameter \( D \) is twice the radius:
\[ D = 2r = 2 \times 10 = 20 \text{ cm} \]
- The diagonal \( d \) of the square can also be expressed in terms of the side length \( s \) of the square using the formula:
\[ d = s\sqrt{2} \]

6. **Set the two expressions for the diagonal equal**:
- Therefore, we have:
\[ s\sqrt{2} = 20 \]

7. **Solve for the side length \( s \)**:
- Dividing both sides by \( \sqrt{2} \):
\[ s = \frac{20}{\sqrt{2}} = 20 \times \frac{\sqrt{2}}{2} = 10\sqrt{2} \text{ cm} \]

### Final Answer:
The length of each side of the square is \( 10\sqrt{2} \) cm.

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