The perimeter of an equilateral triangle is equal to the circumference of a circle. The ratio of their areas is
(Use `pi = (22)/(7)`)

Options

A. `22 : 21sqrt(3)`
B. `21 : 22sqrt(3)`
C. `21 : 22sqrt(2)`
D. `22 : 21sqrt(2)`

Answer

To solve the problem, we need to find the ratio of the areas of an equilateral triangle and a circle, given that their perimeter and circumference are equal.

### Step-by-Step Solution:

1. **Define Variables:**
Let the side length of the equilateral triangle be \( A \) and the radius of the circle be \( R \).

2. **Perimeter of the Equilateral Triangle:**
The perimeter \( P \) of an equilateral triangle is given by:
\[ P = 3A \]

3. **Circumference of the Circle:**
The circumference \( C \) of a circle is given by:
\[ C = 2\pi R \]

4. **Set the Perimeter Equal to the Circumference:**
According to the problem, the perimeter of the triangle is equal to the circumference of the circle:
\[ 3A = 2\pi R \]

5. **Express \( A \) in terms of \( R \):**
Rearranging the equation gives:
\[ A = \frac{2\pi R}{3} \]

6. **Area of the Equilateral Triangle:**
The area \( A_T \) of an equilateral triangle is given by:
\[ A_T = \frac{\sqrt{3}}{4} A^2 \]
Substituting \( A \) from the previous step:
\[ A_T = \frac{\sqrt{3}}{4} \left(\frac{2\pi R}{3}\right)^2 = \frac{\sqrt{3}}{4} \cdot \frac{4\pi^2 R^2}{9} = \frac{\sqrt{3}\pi^2 R^2}{9} \]

7. **Area of the Circle:**
The area \( A_C \) of the circle is given by:
\[ A_C = \pi R^2 \]

8. **Ratio of Areas:**
Now, we need to find the ratio of the area of the triangle to the area of the circle:
\[ \text{Ratio} = \frac{A_T}{A_C} = \frac{\frac{\sqrt{3}\pi^2 R^2}{9}}{\pi R^2} \]
Simplifying this gives:
\[ \text{Ratio} = \frac{\sqrt{3}\pi^2}{9\pi} = \frac{\sqrt{3}\pi}{9} \]

9. **Substituting \( \pi \):**
Using \( \pi = \frac{22}{7} \):
\[ \text{Ratio} = \frac{\sqrt{3} \cdot \frac{22}{7}}{9} = \frac{22\sqrt{3}}{63} \]

10. **Final Ratio:**
To express the ratio in a more recognizable form, we can multiply both the numerator and the denominator by 7:
\[ \text{Final Ratio} = \frac{22\sqrt{3}}{63} \cdot \frac{7}{7} = \frac{154\sqrt{3}}{441} \]
This can be simplified further, but the answer can be expressed in terms of the options given.