A rectangle of maximum area is drawn inside a circle of diameter 5 cm. What is the maximum area of such a rectangle?

Options

A. `25 cm^(2)`
B. `12 cm^(2)`
C. `12.5 cm^(2)`
D. None of these

Answer

To find the maximum area of a rectangle that can be inscribed in a circle with a diameter of 5 cm, we can follow these steps:

### Step-by-Step Solution:

1. **Identify the Diameter of the Circle:**
The diameter of the circle is given as 5 cm.

2. **Calculate the Radius of the Circle:**
The radius (r) of the circle can be calculated using the formula:
\[ r = \frac{\text{Diameter}}{2} = \frac{5 \text{ cm}}{2} = 2.5 \text{ cm} \]

3. **Understand the Relationship between the Rectangle and the Circle:**
The rectangle of maximum area that can be inscribed in a circle is a square. This is because, for a fixed perimeter, a square has the maximum area among all rectangles.

4. **Relate the Diagonal of the Square to the Circle's Diameter:**
The diagonal of the square will be equal to the diameter of the circle. Therefore, if we denote the side length of the square as \( x \), the diagonal \( d \) can be expressed using the Pythagorean theorem:
\[ d = x\sqrt{2} \]
Since the diagonal is equal to the diameter of the circle:
\[ x\sqrt{2} = 5 \text{ cm} \]

5. **Solve for the Side Length \( x \):**
Rearranging the equation gives:
\[ x = \frac{5}{\sqrt{2}} = \frac{5\sqrt{2}}{2} \text{ cm} \]

6. **Calculate the Area of the Square:**
The area \( A \) of the square is given by:
\[ A = x^2 \]
Substituting the value of \( x \):
\[ A = \left(\frac{5\sqrt{2}}{2}\right)^2 = \frac{25 \cdot 2}{4} = \frac{50}{4} = 12.5 \text{ cm}^2 \]

### Final Answer:
The maximum area of the rectangle inscribed in the circle is **12.5 cm²**.

---