The circumferences of two circles are in the ratio 3:4. Find the ratio of their areas.

Options

A. `9:16`
B. `16:9`
C. `6:9`
D. `9:6`

Answer

To find the ratio of the areas of two circles given the ratio of their circumferences, we can follow these steps:

### Step 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.

### Step 2: Set up the ratio of the circumferences.
Given that the circumferences of the two circles are in the ratio \( 3:4 \), we can express this as:
\[ \frac{C_1}{C_2} = \frac{3}{4} \]
Let \( C_1 = 2\pi r_1 \) and \( C_2 = 2\pi r_2 \), where \( r_1 \) and \( r_2 \) are the radii of the first and second circles, respectively.

### Step 3: Substitute the circumference formulas into the ratio.
Substituting the formulas for circumference into the ratio gives us:
\[ \frac{2\pi r_1}{2\pi r_2} = \frac{3}{4} \]
The \( 2\pi \) cancels out:
\[ \frac{r_1}{r_2} = \frac{3}{4} \]

### Step 4: Find the ratio of the radii.
From the above step, we have:
\[ r_1 : r_2 = 3 : 4 \]

### Step 5: Find the ratio of the areas.
The area \( A \) of a circle is given by the formula:
\[ A = \pi r^2 \]
Thus, the areas of the two circles can be expressed as:
\[ A_1 = \pi r_1^2 \quad \text{and} \quad A_2 = \pi r_2^2 \]
Now, we can find the ratio of the areas:
\[ \frac{A_1}{A_2} = \frac{\pi r_1^2}{\pi r_2^2} = \frac{r_1^2}{r_2^2} \]

### Step 6: Substitute the ratio of the radii into the area ratio.
We already found that \( \frac{r_1}{r_2} = \frac{3}{4} \). Therefore:
\[ \frac{r_1^2}{r_2^2} = \left(\frac{3}{4}\right)^2 = \frac{9}{16} \]

### Conclusion: State the final ratio of the areas.
Thus, the ratio of the areas of the two circles is:
\[ A_1 : A_2 = 9 : 16 \]

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