The circumference of a circle le 88 cm. What is the area, in sq.com of a square inscribed in the circle?

(1) 312 (3) 392

(5) 342

(2) 408

(4) 196

To solve this problem, let's first analyze the situation.

1. **Given**:
   - The circumference of the circle is 88 cm.
   - A square is inscribed in the circle, and we need to find its area.

2. **Step 1: Find the radius of the circle**  
   The formula for the circumference of a circle is:
   \[
   C = 2 \pi r
   \]
   where \(r\) is the radius.  
   Given \(C = 88\) cm, we can solve for \(r\):
   \[
   88 = 2 \pi r
   \]
   \[
   r = \frac{88}{2\pi} = \frac{44}{\pi} \approx \frac{44}{3.1416} \approx 14.01 \, \text{cm}
   \]

3. **Step 2: Find the side length of the inscribed square**  
   The diagonal of the inscribed square equals the diameter of the circle. The diameter \(d\) is:
   \[
   d = 2r = 2 \times 14.01 = 28.02 \, \text{cm}
   \]
   The diagonal \(d\) of a square is related to the side length \(s\) by the formula:
   \[
   d = s\sqrt{2}
   \]
   So, solving for \(s\):
   \[
   s = \frac{d}{\sqrt{2}} = \frac{28.02}{\sqrt{2}} \approx \frac{28.02}{1.414} \approx 19.8 \, \text{cm}
   \]

4. **Step 3: Calculate the area of the square**  
   The area \(A\) of the square is:
   \[
   A = s^2 = (19.8)^2 \approx 392.04 \, \text{sq.cm}
   \]

Thus, the area of the inscribed square is approximately **392 sq.cm**.

### Final answer:  
**Option (3): 392**

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Do you need further details or have any questions?

### Related questions:
1. How do you calculate the area of a circle given the circumference?
2. What is the relationship between a circle's radius and an inscribed square's side length?
3. How can you use the Pythagorean theorem to solve for the diagonal of a square?
4. What would the area of a square be if its diagonal was increased by 10%?
5. How does the side length of an inscribed square change with a larger circle?

### Tip:  
Always remember, the diagonal of an inscribed square is equal to the diameter of the circle.

The circumference of a circle is 88 cm. What is the area, in sq. cm, of a square inscribed in the circle?
(1) 312 (3) 392 (5) 342 (2) 408 (4) 196

Discover how to calculate the area of a square inscribed in a circle with a circumference of 88 cm. This problem involves key geometric concepts like the relationship between a circle's radius and an inscribed square's side length, using the Pythagorean theorem and relevant formulas. Suitable for students in Grades 8-10.

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