The ratio of the radius of two circles is 4:5. 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. **Understand the Given Ratio**:
We are given the ratio of the radii of two circles, which is 4:5. Let's denote the radius of the first circle as \( r_1 \) and the radius of the second circle as \( r_2 \).

**Hint**: Write the radii in terms of a variable for easier calculations.

2. **Express the Radii**:
Let the radius of the first circle \( r_1 = 4x \) and the radius of the second circle \( r_2 = 5x \), where \( x \) is a common multiplier.

**Hint**: Use a variable to represent the common factor in the ratio.

3. **Formula for Area of a Circle**:
The area \( A \) of a circle is given by the formula:
\[ A = \pi r^2 \]

4. **Calculate the Areas**:
- For the first circle:
\[ A_1 = \pi (r_1)^2 = \pi (4x)^2 = \pi (16x^2) = 16\pi x^2 \]
- For the second circle:
\[ A_2 = \pi (r_2)^2 = \pi (5x)^2 = \pi (25x^2) = 25\pi x^2 \]

**Hint**: Substitute the expressions for \( r_1 \) and \( r_2 \) into the area formula.

5. **Find the Ratio of the Areas**:
Now, we can find the ratio of the areas \( A_1 \) and \( A_2 \):
\[ \frac{A_1}{A_2} = \frac{16\pi x^2}{25\pi x^2} \]
Here, \( \pi \) and \( x^2 \) cancel out:
\[ \frac{A_1}{A_2} = \frac{16}{25} \]

6. **Express the Result as a Ratio**:
Therefore, the ratio of the areas of the two circles is:
\[ A_1 : A_2 = 16 : 25 \]

### Final Answer:
The ratio of the areas of the two circles is \( 16 : 25 \).