What is the area of a square inscribed in a circle of diameter x cm?

Answer

To find the area of a square inscribed in a circle with a diameter of \( x \) cm, we can follow these steps:

### Step-by-Step Solution:

1. **Understand the relationship between the square and the circle**:
- The square is inscribed in the circle, meaning all four corners of the square touch the circumference of the circle.

2. **Identify the diameter of the circle**:
- Given that the diameter of the circle is \( x \) cm.

3. **Determine the radius of the circle**:
- The radius \( r \) of the circle is half of the diameter.
\[ r = \frac{x}{2} \text{ cm} \]

4. **Relate the side of the square to the radius of the circle**:
- Let the side length of the square be \( a \) cm. The diagonal of the square is equal to the diameter of the circle.
- By the properties of a square, the diagonal \( d \) can be expressed in terms of the side length \( a \) using the Pythagorean theorem:
\[ d = a\sqrt{2} \]
- Since the diagonal of the square is also equal to the diameter of the circle, we have:
\[ a\sqrt{2} = x \]

5. **Solve for the side length \( a \)**:
- Rearranging the equation gives:
\[ a = \frac{x}{\sqrt{2}} \]

6. **Calculate the area of the square**:
- The area \( A \) of the square is given by:
\[ A = a^2 \]
- Substituting the expression for \( a \):
\[ A = \left(\frac{x}{\sqrt{2}}\right)^2 = \frac{x^2}{2} \text{ cm}^2 \]

### Final Answer:
The area of the square inscribed in a circle of diameter \( x \) cm is:
\[ \frac{x^2}{2} \text{ cm}^2 \]