An equilateral triangle with side 6 cm is inscribed in a circle, which in turn is inscribed in a regular hexagon. What is the area of the regular hexagon (in sq. cm)?

Sure, here are the steps to solve the problem:

Step 1: Find the radius of the circle. An equilateral triangle with a side length of 6 cm is inscribed in a circle. The radius of the circle is the same as the height of the triangle. The height of an equilateral triangle can be found using the formula: √3/2 * side. So, the radius of the circle is √3/2 * 6 = 3√3 cm.

Step 2: Find the side length of the hexagon. A regular hexagon inscribed in a circle has the same side length as the radius of the circle. So, the side length of the hexagon is also 3√3 cm.

Step 3: Find the area of the hexagon. The area of a regular hexagon can be found using the formula: 3/2 * √3 * side^2. So, the area of the hexagon is 3/2 * √3 * (3√3)^2 = 3/2 * √3 * 27 = 40.5√3 cm^2.

So, the area of the regular hexagon is 40.5√3 cm^2.