If the circumference of two circles are in the ratio 3:1, then the ratio of their areas will be

Options

A. `3:4 `
B. `9:1`
C. `5:6`
D. `2:3`

Answer

To solve the problem of finding the ratio of the areas of two circles when their circumferences are in the ratio of 3:1, we can follow these steps:

### Step-by-Step Solution:

1. **Understand the relationship between circumference and radius**:
The circumference \( C \) of a circle is given by the formula:
\[ C = 2\pi r \]
where \( r \) is the radius of the circle.

2. **Set up the ratio of circumferences**:
Let the circumferences of the two circles be \( C_1 \) and \( C_2 \). According to the problem, we have:
\[ \frac{C_1}{C_2} = \frac{3}{1} \]

3. **Express the circumferences in terms of their radii**:
For the two circles, using the circumference formula:
\[ C_1 = 2\pi r_1 \quad \text{and} \quad C_2 = 2\pi r_2 \]
Thus, we can write the ratio of the circumferences as:
\[ \frac{2\pi r_1}{2\pi r_2} = \frac{r_1}{r_2} \]

4. **Relate the radii using the circumference ratio**:
From the circumference ratio \( \frac{C_1}{C_2} = \frac{3}{1} \), we can conclude:
\[ \frac{r_1}{r_2} = \frac{3}{1} \]
This means \( r_1 = 3k \) and \( r_2 = k \) for some constant \( k \).

5. **Calculate the areas of the circles**:
The area \( A \) of a circle is given by the formula:
\[ A = \pi r^2 \]
Therefore, the areas of the two circles are:
\[ A_1 = \pi (r_1)^2 = \pi (3k)^2 = 9\pi k^2 \]
\[ A_2 = \pi (r_2)^2 = \pi (k)^2 = \pi k^2 \]

6. **Set up the ratio of the areas**:
Now, we can find the ratio of the areas:
\[ \frac{A_1}{A_2} = \frac{9\pi k^2}{\pi k^2} = \frac{9}{1} \]

7. **Conclusion**:
Thus, the ratio of the areas of the two circles is:
\[ \frac{A_1}{A_2} = 9:1 \]

### Final Answer:
The ratio of the areas of the two circles is \( 9:1 \).

---