The height of an equilateral triangle is `4sqrt(3) cm`. The ratio of the area of its circumcircle to that of its in-circle is

Options

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

Answer

To solve the problem, we need to find the ratio of the area of the circumcircle to that of the incircle of an equilateral triangle with a given height.

### Step-by-step solution:

1. **Identify the height formula for an equilateral triangle**:
The height \( h \) of an equilateral triangle can be expressed in terms of its side length \( a \) as:
\[ h = \frac{\sqrt{3}}{2} a \]

2. **Set the height equal to the given value**:
We are given that the height \( h = 4\sqrt{3} \) cm. Therefore, we can set up the equation:
\[ \frac{\sqrt{3}}{2} a = 4\sqrt{3} \]

3. **Solve for the side length \( a \)**:
To find \( a \), we can multiply both sides by \( 2 \) and then divide by \( \sqrt{3} \):
\[ a = \frac{2 \cdot 4\sqrt{3}}{\sqrt{3}} = 8 \text{ cm} \]

4. **Calculate the circumradius \( R \)**:
The circumradius \( R \) of an equilateral triangle is given by:
\[ R = \frac{a}{\sqrt{3}} \]
Substituting \( a = 8 \):
\[ R = \frac{8}{\sqrt{3}} \text{ cm} \]

5. **Calculate the inradius \( r \)**:
The inradius \( r \) of an equilateral triangle is given by:
\[ r = \frac{a}{2\sqrt{3}} \]
Substituting \( a = 8 \):
\[ r = \frac{8}{2\sqrt{3}} = \frac{4}{\sqrt{3}} \text{ cm} \]

6. **Calculate the area of the circumcircle**:
The area \( A_c \) of the circumcircle is given by:
\[ A_c = \pi R^2 \]
Substituting \( R = \frac{8}{\sqrt{3}} \):
\[ A_c = \pi \left(\frac{8}{\sqrt{3}}\right)^2 = \pi \cdot \frac{64}{3} = \frac{64\pi}{3} \text{ cm}^2 \]

7. **Calculate the area of the incircle**:
The area \( A_i \) of the incircle is given by:
\[ A_i = \pi r^2 \]
Substituting \( r = \frac{4}{\sqrt{3}} \):
\[ A_i = \pi \left(\frac{4}{\sqrt{3}}\right)^2 = \pi \cdot \frac{16}{3} = \frac{16\pi}{3} \text{ cm}^2 \]

8. **Find the ratio of the areas**:
The ratio of the area of the circumcircle to that of the incircle is:
\[ \text{Ratio} = \frac{A_c}{A_i} = \frac{\frac{64\pi}{3}}{\frac{16\pi}{3}} = \frac{64}{16} = 4 \]

### Final Answer:
The ratio of the area of the circumcircle to that of the incircle is \( 4 \).