To find the area of the regular hexagon formed by cutting the corners of an equilateral triangle, we can use the following steps:

1. Find the area of the equilateral triangle:
The area of an equilateral triangle can be calculated using the formula:
Area = (sqrt(3) / 4) * side^2
Given that the side of the equilateral triangle is 6 cm, we can substitute this value into the formula:
Area = (sqrt(3) / 4) * 6^2 = (sqrt(3) / 4) * 36 = 9sqrt(3) cm^2

2. Find the area of a single triangle cut off from the equilateral triangle:
When we cut off the corners of the equilateral triangle, we form 6 congruent triangles. Each of these triangles is equilateral since all sides are equal.
The base of each triangle is equal to the side length of the hexagon, which is also 6 cm.
The height of each triangle can be found by drawing an altitude from one of the vertices to the opposite side. Since the triangle is equilateral, the altitude will also act as a median and a perpendicular bisector.
By drawing the altitude, we form a right-angled triangle with the hypotenuse equal to the side length of the equilateral triangle (6 cm) and one of the legs equal to half of the side length of the hexagon (3 cm).
Using the Pythagorean theorem, we can find the height of the triangle:
height = sqrt(hypotenuse^2 - leg^2) = sqrt(6^2 - 3^2) = sqrt(36 - 9) = sqrt(27) = 3sqrt(3) cm
The area of each triangle can be calculated using the formula:
Area = (sqrt(3) / 4) * base * height = (sqrt(3) / 4) * 6 * 3sqrt(3) = 9sqrt(3) cm^2

3. Find the area of the regular hexagon:
Since the hexagon is formed by 6 congruent triangles, the area of the hexagon is equal to the sum of the areas of these triangles.
Area = 6 * Area of each triangle = 6 * 9sqrt(3) cm^2 = 54sqrt(3) cm^2

Therefore, the area of the regular hexagon formed by cutting the corners of the equilateral triangle is 54sqrt(3) cm^2, which is equivalent to option C.