If the apothem of a regular hexagon is $\mathrm{a}$, find each side of the hexagon.

In the following figure find the ratio of the area of the internal hexagon to that of the external regular hexagon.

In the following figure find the ratio of the shaded triangular area to that of the remaining area of the regular hexagon.

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