The ratio of the area of an equilateral triangle and that of its circumcircle is

Options

A. `2sqrt(3) : 2pi`
B. `4:pi`
C. `3sqrt(3) : 4pi`
D. `7sqrt(2) : 2pi`

Answer

To find the ratio of the area of an equilateral triangle to the area of its circumcircle, we can follow these steps:

### Step 1: Define the side length of the equilateral triangle
Let the side length of the equilateral triangle be denoted as \( a \).

### Step 2: Calculate the area of the equilateral triangle
The formula for the area \( A \) of an equilateral triangle with side length \( a \) is given by:
\[ A = \frac{\sqrt{3}}{4} a^2 \]

### Step 3: Determine the radius of the circumcircle
The radius \( R \) of the circumcircle of an equilateral triangle can be calculated using the formula:
\[ R = \frac{a}{\sqrt{3}} \]

### Step 4: Calculate the area of the circumcircle
The area \( A_c \) of the circumcircle is given by the formula:
\[ A_c = \pi R^2 \]
Substituting the expression for \( R \):
\[ A_c = \pi \left(\frac{a}{\sqrt{3}}\right)^2 = \pi \frac{a^2}{3} \]

### Step 5: Find the ratio of the area of the triangle to the area of the circumcircle
Now, we can find the ratio \( \text{Ratio} \) of the area of the equilateral triangle to the area of the circumcircle:
\[ \text{Ratio} = \frac{A}{A_c} = \frac{\frac{\sqrt{3}}{4} a^2}{\pi \frac{a^2}{3}} \]

### Step 6: Simplify the ratio
We can simplify this expression:
\[ \text{Ratio} = \frac{\frac{\sqrt{3}}{4}}{\frac{\pi}{3}} = \frac{\sqrt{3}}{4} \cdot \frac{3}{\pi} = \frac{3\sqrt{3}}{4\pi} \]

### Final Result
Thus, the ratio of the area of the equilateral triangle to the area of its circumcircle is:
\[ \frac{3\sqrt{3}}{4\pi} \]