If the ratio of areas of two circles is 4:9, then the ratio of their circumference will be :

Options

A. `3:2`
B. `9:4`
C. `2:3`
D. `4:9`

Answer

To solve the problem, we need to find the ratio of the circumferences of two circles given the ratio of their areas is 4:9.

### Step-by-step Solution:

1. **Understanding the Ratio of Areas**:
The ratio of the areas of two circles is given as \( A_1 : A_2 = 4 : 9 \).

2. **Using the Area Formula**:
The area \( A \) of a circle is given by the formula:
\[ A = \pi r^2 \]
where \( r \) is the radius of the circle. Therefore, we can express the areas of the two circles as:
\[ A_1 = \pi r_1^2 \quad \text{and} \quad A_2 = \pi r_2^2 \]

3. **Setting Up the Ratio**:
Using the ratio of the areas, we can write:
\[ \frac{A_1}{A_2} = \frac{\pi r_1^2}{\pi r_2^2} = \frac{r_1^2}{r_2^2} \]
Given \( \frac{A_1}{A_2} = \frac{4}{9} \), we have:
\[ \frac{r_1^2}{r_2^2} = \frac{4}{9} \]

4. **Taking the Square Root**:
To find the ratio of the radii, we take the square root of both sides:
\[ \frac{r_1}{r_2} = \sqrt{\frac{4}{9}} = \frac{2}{3} \]

5. **Finding the Ratio of Circumferences**:
The circumference \( C \) of a circle is given by the formula:
\[ C = 2\pi r \]
Therefore, the circumferences of the two circles can be expressed as:
\[ C_1 = 2\pi r_1 \quad \text{and} \quad C_2 = 2\pi r_2 \]
The ratio of the circumferences is:
\[ \frac{C_1}{C_2} = \frac{2\pi r_1}{2\pi r_2} = \frac{r_1}{r_2} \]

6. **Substituting the Ratio of Radii**:
We already found that:
\[ \frac{r_1}{r_2} = \frac{2}{3} \]
Therefore, the ratio of the circumferences is:
\[ \frac{C_1}{C_2} = \frac{2}{3} \]

### Final Answer:
The ratio of the circumferences of the two circles is \( 2 : 3 \).

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