Inner hexagon:

You can see that the radius of the circle is the length of each equilateral triangle, so evaluating the area of each equilateral triangle using Heron's Formula:

So the area of the hexagon is

Outer hexagon:

You can see that the radius of the circle is the height of each equilateral triangle, and splits each equilateral triangle into two triangles with the sides in the ratio . So the if the height of each triangle is , then the base is .

Therefore the area of each triangle is , and therefore the area of the hexagon is