The areas of two circles are in the ratio 25:36. Find the ratio of their circumferences.

Options

A. `6:5`
B. `5:6`
C. `7:6`
D. `6:7`

Answer

To solve the problem of finding the ratio of the circumferences of two circles given that their areas are in the ratio 25:36, 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:
\[ \frac{A_1}{A_2} = \frac{25}{36} \]
This can be expressed using the formula for area:
\[ \frac{\pi r_1^2}{\pi r_2^2} = \frac{25}{36} \]
The \( \pi \) cancels out, so we have:
\[ \frac{r_1^2}{r_2^2} = \frac{25}{36} \]

### Step 3: Take the square root of both sides
To find the ratio of the radii, we take the square root of both sides:
\[ \frac{r_1}{r_2} = \sqrt{\frac{25}{36}} = \frac{\sqrt{25}}{\sqrt{36}} = \frac{5}{6} \]

### Step 4: Use the ratio of the radii to find the ratio of the circumferences
The circumference \( C \) of a circle is given by the formula:
\[ C = 2\pi r \]
Thus, the circumferences of the two circles can be expressed as:
\[ C_1 = 2\pi r_1 \quad \text{and} \quad C_2 = 2\pi r_2 \]
Now, we can find the ratio of the circumferences:
\[ \frac{C_1}{C_2} = \frac{2\pi r_1}{2\pi r_2} = \frac{r_1}{r_2} \]
Substituting the ratio we found:
\[ \frac{C_1}{C_2} = \frac{5}{6} \]

### Final Answer
The ratio of the circumferences of the two circles is:
\[ \frac{C_1}{C_2} = \frac{5}{6} \]

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