What is the ratio of the area of circumcircle of equilateral triangle to the area of square with the same side length as the equilateral triangle ?

Options

A. `pi : 3`
B. `pi : sqrt(3) `
C. `sqrt(3):2`
D. none of these

Answer

To find the ratio of the area of the circumcircle of an equilateral triangle to the area of a square with the same side length as the equilateral triangle, we can follow these steps:

### Step-by-Step Solution:

1. **Identify the side length of the equilateral triangle**:
Let the side length of the equilateral triangle be \( a \).

2. **Calculate the circumradius of the equilateral triangle**:
The formula for the circumradius \( R \) of an equilateral triangle is given by:
\[ R = \frac{a}{\sqrt{3}} \]

3. **Calculate the area of the circumcircle**:
The area \( A_c \) of the circumcircle can be calculated using the formula:
\[ A_c = \pi R^2 \]
Substituting the value of \( R \):
\[ A_c = \pi \left(\frac{a}{\sqrt{3}}\right)^2 = \pi \frac{a^2}{3} \]

4. **Calculate the area of the square**:
The area \( A_s \) of the square with side length \( a \) is:
\[ A_s = a^2 \]

5. **Find the ratio of the area of the circumcircle to the area of the square**:
The ratio \( \text{Ratio} \) is given by:
\[ \text{Ratio} = \frac{A_c}{A_s} = \frac{\pi \frac{a^2}{3}}{a^2} \]
Simplifying this gives:
\[ \text{Ratio} = \frac{\pi}{3} \]

6. **Conclusion**:
Therefore, the ratio of the area of the circumcircle of the equilateral triangle to the area of the square with the same side length is:
\[ \text{Ratio} = \frac{\pi}{3} \]

### Final Answer:
The ratio is \( \pi : 3 \).

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