Consider a circle with radius r. How do i work out the area of the circumscribe hexagon? I have managed to work out the area of the inner to be 3asqrt{r^2-a^2}.
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