In the following figure a hexagram consists of a hexagon and six equilateral triangles outside the hexagon. If a circle is inscribed in the hexagon and the area of the inscribed circle is $3\mathrm{\pi }$ $\mathrm{sq}.$ $\mathrm{cm}$, find the area of the circumscribed circle of the regular hexagram.

In the following hexagonal ring the area of shaded region is $3$ times the area of the un-shaded region. Find the ratio of perimeter of the outer hexagon to that of the inner hexagon.

"Find the value of angle $x$ in a regular octagon as shown below

In the following figure the regular octagon has some shaded area and some unshaded area. Find the ratio of the shaded area to the unshaded area.