An equilateral triangle of side 6 cm has its corners cut off to form a regular hexagon. Area (in `cm^(2)`) of this regular hexagon will be

Options

A. `3 sqrt(3) `
B. `3 sqrt(6)`
C. `6 sqrt(3) `
D. `(5 sqrt(3))/(2)`

Answer

To find the area of the regular hexagon formed by cutting off the corners of an equilateral triangle with a side length of 6 cm, we can follow these steps:

### Step-by-Step Solution:

1. **Calculate the Area of the Equilateral Triangle:**
The formula for the area \( A \) of an equilateral triangle with side length \( a \) is given by:
\[ A = \frac{\sqrt{3}}{4} a^2 \]
Substituting \( a = 6 \) cm:
\[ A = \frac{\sqrt{3}}{4} \times 6^2 = \frac{\sqrt{3}}{4} \times 36 = 9\sqrt{3} \text{ cm}^2 \]

2. **Determine the Area of the Corners Cut Off:**
When the corners of the triangle are cut off to form a regular hexagon, each corner removed is an equilateral triangle with a side length of \( 2 \) cm (since the original triangle's side length is 6 cm, and each corner cut removes 2 cm from each side).
The area of one small equilateral triangle (corner) is:
\[ A_{\text{corner}} = \frac{\sqrt{3}}{4} \times 2^2 = \frac{\sqrt{3}}{4} \times 4 = \sqrt{3} \text{ cm}^2 \]
Since there are 3 corners cut off:
\[ A_{\text{total corners}} = 3 \times \sqrt{3} = 3\sqrt{3} \text{ cm}^2 \]

3. **Calculate the Area of the Hexagon:**
The area of the hexagon can be found by subtracting the total area of the corners from the area of the original triangle:
\[ A_{\text{hexagon}} = A_{\text{triangle}} - A_{\text{total corners}} = 9\sqrt{3} - 3\sqrt{3} = 6\sqrt{3} \text{ cm}^2 \]

### Final Answer:
The area of the regular hexagon is:
\[ \boxed{6\sqrt{3} \text{ cm}^2} \]