If the ratio of the radii of two circles is 2:3, find the ratio of their areas.

Answer

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

### Step-by-Step Solution:

1. **Identify the Radii Ratio**:
We are given that the ratio of the radii of two circles is 2:3. Let's denote the radius of the first circle as \( r_1 \) and the radius of the second circle as \( r_2 \). Therefore, we can write:
\[ \frac{r_1}{r_2} = \frac{2}{3} \]

2. **Write the Area Formula**:
The area \( A \) of a circle is given by the formula:
\[ A = \pi r^2 \]
So, the area of the first circle \( A_1 \) and the area of the second circle \( A_2 \) can be expressed as:
\[ A_1 = \pi r_1^2 \quad \text{and} \quad A_2 = \pi r_2^2 \]

3. **Find the Ratio of the Areas**:
To find the ratio of the areas \( \frac{A_1}{A_2} \), we substitute the area formulas:
\[ \frac{A_1}{A_2} = \frac{\pi r_1^2}{\pi r_2^2} \]
We can cancel \( \pi \) from the numerator and the denominator:
\[ \frac{A_1}{A_2} = \frac{r_1^2}{r_2^2} \]

4. **Substitute the Radii Ratio**:
Now we substitute the ratio of the radii into the area ratio:
\[ \frac{A_1}{A_2} = \left(\frac{r_1}{r_2}\right)^2 = \left(\frac{2}{3}\right)^2 \]

5. **Calculate the Square of the Ratio**:
Now we calculate the square:
\[ \left(\frac{2}{3}\right)^2 = \frac{2^2}{3^2} = \frac{4}{9} \]

6. **Final Ratio of the Areas**:
Therefore, the ratio of the areas of the two circles is:
\[ \frac{A_1}{A_2} = \frac{4}{9} \]

### Final Answer:
The ratio of the areas of the two circles is \( 4:9 \).

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