Find the ratio of circumference 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 circumference of two circles whose areas are in the ratio of 16:25, we can follow these steps:

### Step 1: Understand the relationship between the area and radius of a circle.
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 radius of the first circle be \( r_1 \) and the radius of the second circle be \( r_2 \). According to the problem, the areas of the two circles are in the ratio:
\[ \frac{A_1}{A_2} = \frac{16}{25} \]
Substituting the area formula, we have:
\[ \frac{\pi r_1^2}{\pi r_2^2} = \frac{16}{25} \]
The \( \pi \) cancels out, leading to:
\[ \frac{r_1^2}{r_2^2} = \frac{16}{25} \]

### Step 3: Take the square root to find the ratio of the radii.
Taking the square root of both sides gives:
\[ \frac{r_1}{r_2} = \sqrt{\frac{16}{25}} = \frac{4}{5} \]

### Step 4: Find the ratio of the circumferences.
The circumference \( C \) of a circle is given by the formula:
\[ C = 2\pi r \]
Thus, the ratio of the circumferences of the two circles is:
\[ \frac{C_1}{C_2} = \frac{2\pi r_1}{2\pi r_2} \]
The \( 2\pi \) cancels out, leading to:
\[ \frac{C_1}{C_2} = \frac{r_1}{r_2} \]

### Step 5: Substitute the ratio of the radii.
From Step 3, we found that:
\[ \frac{r_1}{r_2} = \frac{4}{5} \]
Thus, the ratio of the circumferences is:
\[ \frac{C_1}{C_2} = \frac{4}{5} \]

### Final Answer:
The ratio of the circumference of the two circles is \( 4:5 \).

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