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. `3sqrt(3)`
B. `3sqrt(6)`
C. `6sqrt(3)`
D. `(5sqrt(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. **Understand the Geometry**:
- We start with an equilateral triangle ABC where each side is 6 cm.
- When we cut off the corners of the triangle, we form a regular hexagon.

2. **Determine the Side Length of the Hexagon**:
- Each corner cut off creates two new vertices of the hexagon. Since the triangle is equilateral, the cuts are equal.
- The original triangle has a side length of 6 cm. Each corner cut off reduces the length of each side of the triangle by 2 cm (1 cm from each end).
- Therefore, the side length of the hexagon is:
\[ \text{Side length of hexagon} = \frac{6 \text{ cm}}{3} = 2 \text{ cm} \]

3. **Calculate the Area of the Hexagon**:
- The formula for the area \( A \) of a regular hexagon with side length \( a \) is given by:
\[ A = \frac{3\sqrt{3}}{2} a^2 \]
- Substituting \( a = 2 \text{ cm} \):
\[ A = \frac{3\sqrt{3}}{2} (2)^2 \]
\[ A = \frac{3\sqrt{3}}{2} \times 4 \]
\[ A = 6\sqrt{3} \text{ cm}^2 \]

4. **Final Answer**:
- The area of the regular hexagon is:
\[ \text{Area} = 6\sqrt{3} \text{ cm}^2 \]