In the following diagram, as shown below, a regular hexagon of side $3 \mathrm{cm}$ inscribes $7$ smaller regular hexagons. Find the area of the shaded region (of the triangles) within the larger hexagon.

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