If the ratio of areas of two squares is 225 : 256, then the ratio of their perimeter is :

Options

A. `225:256`
B. `256 : 225`
C. `15 : 16`
D. `16 : 15`

Answer

To solve the problem of finding the ratio of the perimeters of two squares when the ratio of their areas is given as 225:256, we can follow these steps:

### Step-by-Step Solution:

1. **Understand the relationship between area and side length of a square**:
The area \( A \) of a square is given by the formula \( A = s^2 \), where \( s \) is the length of a side of the square.

2. **Set up the ratio of the areas**:
Given that the ratio of the areas of two squares is \( 225:256 \), we can express this as:
\[ \frac{A_1}{A_2} = \frac{225}{256} \]

3. **Take the square root of the ratio of areas**:
To find the ratio of the side lengths, we take the square root of both sides:
\[ \frac{s_1}{s_2} = \sqrt{\frac{225}{256}} = \frac{\sqrt{225}}{\sqrt{256}} = \frac{15}{16} \]

4. **Calculate the ratio of the perimeters**:
The perimeter \( P \) of a square is given by the formula \( P = 4s \). Therefore, the ratio of the perimeters of the two squares can be calculated as follows:
\[ \frac{P_1}{P_2} = \frac{4s_1}{4s_2} = \frac{s_1}{s_2} \]
Since we already found that \( \frac{s_1}{s_2} = \frac{15}{16} \), we can conclude:
\[ \frac{P_1}{P_2} = \frac{15}{16} \]

5. **Final Answer**:
The ratio of the perimeters of the two squares is \( 15:16 \).