The ratio of the areas of the circumcircle and the incircle of an equilateral triangle is

Options

A. `2:1`
B. `4:1`
C. `8:1`
D. `3:2`

Answer

To find the ratio of the areas of the circumcircle and the incircle of an equilateral triangle, we can follow these steps:

### Step 1: Understand the Radii of the Circles
For an equilateral triangle with side length \( a \):
- The radius of the circumcircle \( R \) is given by the formula:
\[ R = \frac{a}{\sqrt{3}} \]
- The radius of the incircle \( r \) is given by the formula:
\[ r = \frac{a}{2\sqrt{3}} \]

### Step 2: Calculate the Area of the Circumcircle
The area \( A_c \) of the circumcircle is calculated using the formula for the area of a circle:
\[ A_c = \pi R^2 \]
Substituting the value of \( R \):
\[ A_c = \pi \left(\frac{a}{\sqrt{3}}\right)^2 = \pi \frac{a^2}{3} \]

### Step 3: Calculate the Area of the Incircle
The area \( A_i \) of the incircle is also calculated using the area formula:
\[ A_i = \pi r^2 \]
Substituting the value of \( r \):
\[ A_i = \pi \left(\frac{a}{2\sqrt{3}}\right)^2 = \pi \frac{a^2}{12} \]

### Step 4: Find the Ratio of the Areas
Now, we need to find the ratio of the area of the circumcircle to the area of the incircle:
\[ \text{Ratio} = \frac{A_c}{A_i} = \frac{\pi \frac{a^2}{3}}{\pi \frac{a^2}{12}} \]
The \( \pi \) and \( a^2 \) terms cancel out:
\[ \text{Ratio} = \frac{\frac{1}{3}}{\frac{1}{12}} = \frac{12}{3} = 4 \]

### Step 5: Final Result
Thus, the ratio of the areas of the circumcircle and the incircle of an equilateral triangle is:
\[ \text{Ratio} = 4:1 \]