If ratio of area of a circle to the area of an equilateral triangle is `16pi :sqrt3,` then what is the ratio of radius of the circle to the length of the side of equilateral triangle?

Options

A. `2:3`
B. `2:1`
C. `1:2`
D. `3:2`

Answer

To solve the problem, we need to find the ratio of the radius of a circle to the length of the side of an equilateral triangle, given that the ratio of the area of the circle to the area of the triangle is \(16\pi : \sqrt{3}\).

### Step-by-Step Solution:

1. **Understand the Areas**:
- The area of a circle is given by the formula:
\[ \text{Area of Circle} = \pi r^2 \]
- The area of an equilateral triangle with side length \(a\) is given by the formula:
\[ \text{Area of Triangle} = \frac{\sqrt{3}}{4} a^2 \]

2. **Set Up the Ratio**:
- According to the problem, the ratio of the area of the circle to the area of the triangle is:
\[ \frac{\pi r^2}{\frac{\sqrt{3}}{4} a^2} = \frac{16\pi}{\sqrt{3}} \]

3. **Cross-Multiply to Eliminate the Fraction**:
- Cross-multiplying gives:
\[ \pi r^2 \cdot \sqrt{3} = 16\pi \cdot \frac{\sqrt{3}}{4} a^2 \]
- Simplifying the right side:
\[ \pi r^2 \cdot \sqrt{3} = 4\sqrt{3} \pi a^2 \]

4. **Cancel Common Terms**:
- We can cancel \(\pi\) from both sides (assuming \(\pi \neq 0\)):
\[ r^2 \cdot \sqrt{3} = 4\sqrt{3} a^2 \]

5. **Divide Both Sides by \(\sqrt{3}\)**:
- Dividing both sides by \(\sqrt{3}\) gives:
\[ r^2 = 4a^2 \]

6. **Take the Square Root**:
- Taking the square root of both sides results in:
\[ r = 2a \]

7. **Find the Ratio**:
- The ratio of the radius of the circle \(r\) to the length of the side of the triangle \(a\) is:
\[ \frac{r}{a} = \frac{2a}{a} = 2 \]
- Therefore, the ratio is:
\[ r : a = 2 : 1 \]

### Final Answer:
The ratio of the radius of the circle to the length of the side of the equilateral triangle is \(2 : 1\).