The radii of two circles are in the ratio ` 2:5.` What is 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**:
Let the radius of the first circle be \( r_1 \) and the radius of the second circle be \( r_2 \). According to the problem, the ratio of the radii is given as:
\[ r_1 : r_2 = 2 : 5 \]

2. **Express Radii in Terms of a Variable**:
We can express the radii in terms of a variable \( k \):
\[ r_1 = 2k \quad \text{and} \quad r_2 = 5k \]
where \( k \) is a common multiplier.

3. **Use the Area Formula**:
The area \( A \) of a circle is given by the formula:
\[ A = \pi r^2 \]
Therefore, the areas of the two circles can be expressed as:
\[ A_1 = \pi (r_1)^2 = \pi (2k)^2 = \pi \cdot 4k^2 \]
\[ A_2 = \pi (r_2)^2 = \pi (5k)^2 = \pi \cdot 25k^2 \]

4. **Find the Ratio of the Areas**:
Now, we can find the ratio of the areas \( A_1 \) and \( A_2 \):
\[ \text{Ratio of Areas} = \frac{A_1}{A_2} = \frac{\pi \cdot 4k^2}{\pi \cdot 25k^2} \]
The \( \pi \) and \( k^2 \) terms cancel out:
\[ \frac{A_1}{A_2} = \frac{4}{25} \]

5. **Express the Ratio**:
Thus, the ratio of the areas of the two circles is:
\[ A_1 : A_2 = 4 : 25 \]

### Final Answer:
The ratio of the areas of the two circles is \( 4 : 25 \).

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