Question

An equilateral triangle of side 6 cm has its corners cut-off to form a regular hexagon. The area of the regular hexagon is:

• A. \( 2\sqrt{3} \, {cm}^2 \)
• B. \( 3\sqrt{2} \, {cm}^2 \)
• C. \( 6\sqrt{3} \, {cm}^2 \)
• D. \( 3\sqrt{6} \, {cm}^2 \)

Hint:

For a regular hexagon inscribed inside an equilateral triangle, the area of the hexagon can be calculated using the formula \( \frac{3\sqrt{3}}{2} s^2 \), where \( s \) is the side length of the hexagon.

Answer 1

The Correct Option is C

Step 1: Understanding the geometry of the problem.
The equilateral triangle has a side length of 6 cm. When we cut off the corners, we form a regular hexagon inside the triangle. The regular hexagon is inscribed in the triangle, and its sides are determined by cutting off the corners of the equilateral triangle.

Step 2: Area of regular hexagon.
To find the area of the regular hexagon, we use the formula:

Ahexagon = (3√3/2) s2

where s is the side length of the hexagon.
Since the hexagon is inscribed inside the equilateral triangle, the side of the hexagon corresponds to the side of the smaller triangles that are cut off from the corners.

Step 3: Side length of the hexagon.
For a regular hexagon inscribed inside an equilateral triangle, the side length of the hexagon is the same as the side length of the smaller triangles formed by cutting the corners. This side length is equal to s/2, where s is the side length of the equilateral triangle.
So, the side length of the hexagon is:

shexagon = 6 cm

Step 4: Calculate the area of the hexagon.
Now, we calculate the area of the regular hexagon:

Ahexagon = (3√3/2) (6)2 = (3√3/2) × 36 = 6√3 cm2

Step 5: Conclusion.
The area of the regular hexagon formed by cutting off the corners of the equilateral triangle is 6√3 cm².
Thus, the correct answer is (3) 6√3 cm².