We are told the radius is 6 inches, this distance refers to the distance from the center to a vertice of the hexagon. A regular hexagon is made up of 6 equilateral triangles. The sides of these triangles inside the hexagon are 6 inches. Since these are all equilateral, all sides of the triangle of 6 inches making the 6 outer sides of the hexagon also equal to 6 inches.
We can determine the total area of the hexagon if we know the area of one of the triangles. The area of a triangle is found using the following formula:
A = 1/2 (b x h)
b = the outsider of the hexagon
h = distance from center to the bisection of a side.
We can solve for h by using the pythagorean theorem. We know two sides of the right triangle formed. The adjacent side is 3 inches (half of 6 since the side was bisected) and the hypotenuse is 6 inches.
3² + h² = 6²
h² = 36 - 9
h² = 27
h = √27
h = 3√3
Solving for the area of one triangle is as follows:
A = (1/2)(bh)
A = (1/2)(6)(3√3)
A = 9√3
The total area of the hexagon will be the total of all 6 equilateral triangles.
A = 6(9√3)
A = 54√3
The area of the hexagon is 54√3 or 93.5 in².