The area of the largest possible square inscribed in a circle of unit radius ( in square unit ) is :

Options

A. 3
B. 4
C. `2 sqrt(3) pi `
D. 2

Answer

To find the area of the largest possible square inscribed in a circle of unit radius, we can follow these steps:

### Step-by-Step Solution:

1. **Understand the Circle and Square Relationship**:
- A square inscribed in a circle means that all four corners (vertices) of the square touch the circumference of the circle.

2. **Identify the Radius and Diameter**:
- Given that the radius of the circle is 1 unit, we can calculate the diameter of the circle.
- The diameter \( D \) is given by the formula:
\[ D = 2 \times \text{radius} = 2 \times 1 = 2 \text{ units} \]

3. **Relate the Square's Diagonal to the Circle's Diameter**:
- The diagonal of the square will be equal to the diameter of the circle.
- Therefore, the diagonal \( d \) of the square is:
\[ d = 2 \text{ units} \]

4. **Use the Diagonal to Find the Side Length of the Square**:
- Let the side length of the square be \( x \).
- The relationship between the side length \( x \) and the diagonal \( d \) of a square is given by the formula:
\[ d = x\sqrt{2} \]
- Setting the diagonal equal to the diameter, we have:
\[ x\sqrt{2} = 2 \]

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

6. **Calculate the Area of the Square**:
- The area \( A \) of a square is given by the formula:
\[ A = x^2 \]
- Substituting the value of \( x \):
\[ A = (\sqrt{2})^2 = 2 \text{ square units} \]

### Final Answer:
The area of the largest possible square inscribed in a circle of unit radius is **2 square units**.

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