A square is circumscribed about a circle which is turn circumscribes another square. The ratio of the area of outer square to the inner square

Options

A. `1 : 2`
B. `2 : 1`
C. `1 : 3`
D. `1 : 1`

Answer

To solve the problem, we need to find the ratio of the area of the outer square to the area of the inner square. Let's denote the side length of the inner square as \( a \) and the side length of the outer square as \( b \).

### Step-by-Step Solution:

1. **Understanding the Geometry**:
- The inner square is inscribed in a circle, which means the vertices of the inner square touch the circle.
- The outer square is circumscribed around the same circle, meaning the circle touches the midpoints of the sides of the outer square.

2. **Finding the Diameter of the Circle**:
- The diagonal of the inner square can be calculated using the Pythagorean theorem. For a square with side length \( a \), the diagonal \( d \) is given by:
\[ d = a\sqrt{2} \]
- This diagonal is also the diameter of the circle.

3. **Relating the Outer Square to the Circle**:
- The diameter of the circle is equal to the side length of the outer square \( b \). Therefore, we have:
\[ b = d = a\sqrt{2} \]

4. **Calculating the Areas**:
- The area of the inner square \( A_{inner} \) is:
\[ A_{inner} = a^2 \]
- The area of the outer square \( A_{outer} \) is:
\[ A_{outer} = b^2 = (a\sqrt{2})^2 = 2a^2 \]

5. **Finding the Ratio of Areas**:
- The ratio of the area of the outer square to the area of the inner square is:
\[ \text{Ratio} = \frac{A_{outer}}{A_{inner}} = \frac{2a^2}{a^2} = 2 \]

### Final Answer:
The ratio of the area of the outer square to the area of the inner square is \( 2:1 \).