Find the ratio of circumferences of two circles, whose areas are in the ratio of 16 : 25.

Options

A. ` 16 : 25`
B. `4 : 5`
C. `5 : 4`
D. `25 : 16`

Answer

To find the ratio of the circumferences of two circles whose areas are in the ratio of 16:25, we can follow these steps:

### Step 1: Understand the relationship between area and radius
The area \( A \) of a circle is given by the formula:
\[ A = \pi r^2 \]
where \( r \) is the radius of the circle.

### Step 2: Set up the ratio of the areas
Let the areas of the two circles be \( A_1 \) and \( A_2 \). According to the problem, we have:
\[ \frac{A_1}{A_2} = \frac{16}{25} \]

### Step 3: Express the areas in terms of the radii
Using the area formula, we can express the areas in terms of the radii \( r_1 \) and \( r_2 \):
\[ \frac{\pi r_1^2}{\pi r_2^2} = \frac{16}{25} \]
The \( \pi \) cancels out:
\[ \frac{r_1^2}{r_2^2} = \frac{16}{25} \]

### Step 4: Take the square root of both sides
To find the ratio of the radii, we take the square root of both sides:
\[ \frac{r_1}{r_2} = \sqrt{\frac{16}{25}} = \frac{4}{5} \]

### Step 5: Find the ratio of the circumferences
The circumference \( C \) of a circle is given by the formula:
\[ C = 2\pi r \]
Let \( C_1 \) and \( C_2 \) be the circumferences of the first and second circles, respectively. Then:
\[ \frac{C_1}{C_2} = \frac{2\pi r_1}{2\pi r_2} = \frac{r_1}{r_2} \]
Substituting the ratio of the radii we found:
\[ \frac{C_1}{C_2} = \frac{4}{5} \]

### Final Answer
Thus, the ratio of the circumferences of the two circles is:
\[ \boxed{4:5} \]

---