The diameters of two circles are the side of a square and the diagonal of the square. The ratio of the areas of the smaller circle and the larger circle is

Options

A. `1 : 4`
B. `a//2 : V3`
C. `1 : a//2`
D. `1 : 2`

Answer

To solve the problem, we need to find the ratio of the areas of two circles, where the diameters of the circles are the side of a square and the diagonal of the square.

### Step-by-Step Solution:

1. **Define the side of the square**:
Let the side of the square be \( a \) cm.

**Hint**: Remember that the side of the square is the basic unit we will use to find the diameters of the circles.

2. **Calculate the diagonal of the square**:
The diagonal \( d \) of a square can be calculated using the formula:
\[ d = a \sqrt{2} \]

**Hint**: Use the Pythagorean theorem to derive the formula for the diagonal of a square.

3. **Determine the diameters of the circles**:
- The diameter of the smaller circle is equal to the side of the square, which is \( a \).
- The diameter of the larger circle is equal to the diagonal of the square, which is \( a \sqrt{2} \).

**Hint**: Keep in mind that the diameter is twice the radius.

4. **Calculate the radii of the circles**:
- The radius \( r_1 \) of the smaller circle is:
\[ r_1 = \frac{a}{2} \]
- The radius \( r_2 \) of the larger circle is:
\[ r_2 = \frac{a \sqrt{2}}{2} \]

**Hint**: Remember to divide the diameter by 2 to find the radius.

5. **Find the areas of the circles**:
The area \( A \) of a circle is given by the formula:
\[ A = \pi r^2 \]
- The area of the smaller circle \( A_1 \) is:
\[ A_1 = \pi \left(\frac{a}{2}\right)^2 = \pi \frac{a^2}{4} \]
- The area of the larger circle \( A_2 \) is:
\[ A_2 = \pi \left(\frac{a \sqrt{2}}{2}\right)^2 = \pi \frac{2a^2}{4} = \frac{\pi a^2}{2} \]

**Hint**: Substitute the radius into the area formula carefully.

6. **Calculate the ratio of the areas**:
The ratio of the area of the smaller circle to the area of the larger circle is:
\[ \text{Ratio} = \frac{A_1}{A_2} = \frac{\frac{\pi a^2}{4}}{\frac{\pi a^2}{2}} = \frac{1/4}{1/2} = \frac{1}{2} \]

**Hint**: When dividing fractions, multiply by the reciprocal of the denominator.

### Final Answer:
The ratio of the areas of the smaller circle to the larger circle is \( \frac{1}{2} \).