When all the diagonals of the regular hexagon are drawn in, you should notice that six congruent equilateral triangles are created.

Thus, we know that the given line segment, also known as an apothem, acts as the height of an equilateral triangle. The apothem also creates two congruent \(\displaystyle 30-60-90\) triangles.

Thus, we can use the ratio of the side lengths to find the length of the base of one of the \(\displaystyle 30-60-90\) triangles.

Recall that the ratios of the side lengths in a \(\displaystyle 30-60-90\) triangle are \(\displaystyle 1:\sqrt3:2\).

We can then set up the following equation:

\(\displaystyle \frac{14}{\sqrt3}=\frac{\text{base}}{1}\)

The length of the base is then \(\displaystyle \frac{30}{\sqrt3}=10\sqrt3\).

Now, the length of this base is half the length of a side of the hexagon. Multiply the base by two to find the length of a side:

\(\displaystyle \text{side}=2(\text{base})\)

\(\displaystyle \text{side}=2(10\sqrt3)=20\sqrt3\)

Next, recall how to find the perimeter of a regular hexagon:

\(\displaystyle \text{Perimeter}=6(\text{side})\)

Plug in the side length to find the perimeter:

\(\displaystyle \text{Perimeter}=6(20\sqrt3)=120\sqrt3\)

When all the diagonals of the regular hexagon are drawn in, you should notice that six congruent equilateral triangles are created.

Thus, we know that the given line segment, also known as an apothem, acts as the height of an equilateral triangle. The apothem also creates two congruent \(\displaystyle 30-60-90\) triangles.

Thus, we can use the ratio of the side lengths to find the length of the base of one of the \(\displaystyle 30-60-90\) triangles.

Recall that the ratios of the side lengths in a \(\displaystyle 30-60-90\) triangle are \(\displaystyle 1:\sqrt3:2\).

We can then set up the following equation:

\(\displaystyle \frac{14}{\sqrt3}=\frac{\text{base}}{1}\)

The length of the base is then \(\displaystyle \frac{2}{\sqrt3}=\frac{2\sqrt3}{3}\).

Now, the length of this base is half the length of a side of the hexagon. Multiply the base by two to find the length of a side:

\(\displaystyle \text{side}=2(\text{base})\)

\(\displaystyle \text{side}=2(\frac{2\sqrt3}{3})=\frac{4\sqrt3}{3}\)

Next, recall how to find the perimeter of a regular hexagon:

\(\displaystyle \text{Perimeter}=6(\text{side})\)

Plug in the side length to find the perimeter:

\(\displaystyle \text{Perimeter}=6(\frac{4\sqrt3}{3})=8\sqrt3\)

When all the diagonals of the regular hexagon are drawn in, you should notice that six congruent equilateral triangles are created.

Thus, we know that the given line segment, also known as an apothem, acts as the height of an equilateral triangle. The apothem also creates two congruent \(\displaystyle 30-60-90\) triangles.

Thus, we can use the ratio of the side lengths to find the length of the base of one of the \(\displaystyle 30-60-90\) triangles.

Recall that the ratios of the side lengths in a \(\displaystyle 30-60-90\) triangle are \(\displaystyle 1:\sqrt3:2\).

We can then set up the following equation:

\(\displaystyle \frac{14}{\sqrt3}=\frac{\text{base}}{1}\)

The length of the base is then \(\displaystyle \frac{27}{\sqrt3}=9\sqrt3\).

Now, the length of this base is half the length of a side of the hexagon. Multiply the base by two to find the length of a side:

\(\displaystyle \text{side}=2(\text{base})\)

\(\displaystyle \text{side}=2(9\sqrt3)=18\sqrt3\)

Next, recall how to find the perimeter of a regular hexagon:

\(\displaystyle \text{Perimeter}=6(\text{side})\)

Plug in the side length to find the perimeter:

\(\displaystyle \text{Perimeter}=6(18\sqrt3)=108\sqrt3\)

When all the diagonals of the regular hexagon are drawn in, you should notice that six congruent equilateral triangles are created.

Thus, we know that the given line segment, also known as an apothem, acts as the height of an equilateral triangle. The apothem also creates two congruent \(\displaystyle 30-60-90\) triangles.

Thus, we can use the ratio of the side lengths to find the length of the base of one of the \(\displaystyle 30-60-90\) triangles.

Recall that the ratios of the side lengths in a \(\displaystyle 30-60-90\) triangle are \(\displaystyle 1:\sqrt3:2\).

We can then set up the following equation:

\(\displaystyle \frac{14}{\sqrt3}=\frac{\text{base}}{1}\)

The length of the base is then \(\displaystyle \frac{32}{\sqrt3}=\frac{32\sqrt3}{3}\).

Now, the length of this base is half the length of a side of the hexagon. Multiply the base by two to find the length of a side:

\(\displaystyle \text{side}=2(\text{base})\)

\(\displaystyle \text{side}=2(\frac{32\sqrt3}{3})=\frac{64\sqrt3}{3}\)

Next, recall how to find the perimeter of a regular hexagon:

\(\displaystyle \text{Perimeter}=6(\text{side})\)

Plug in the side length to find the perimeter:

\(\displaystyle \text{Perimeter}=6(\frac{64\sqrt3}{3})=128\sqrt3\)

When all the diagonals of the regular hexagon are drawn in, you should notice that six congruent equilateral triangles are created.

Thus, we know that the given line segment, also known as an apothem, acts as the height of an equilateral triangle. The apothem also creates two congruent \(\displaystyle 30-60-90\) triangles.

Thus, we can use the ratio of the side lengths to find the length of the base of one of the \(\displaystyle 30-60-90\) triangles.

Recall that the ratios of the side lengths in a \(\displaystyle 30-60-90\) triangle are \(\displaystyle 1:\sqrt3:2\).

We can then set up the following equation:

\(\displaystyle \frac{14}{\sqrt3}=\frac{\text{base}}{1}\)

The length of the base is then \(\displaystyle \frac{45}{\sqrt3}=15\sqrt3\).

Now, the length of this base is half the length of a side of the hexagon. Multiply the base by two to find the length of a side:

\(\displaystyle \text{side}=2(\text{base})\)

\(\displaystyle \text{side}=2(15\sqrt3)=30\sqrt3\)

Next, recall how to find the perimeter of a regular hexagon:

\(\displaystyle \text{Perimeter}=6(\text{side})\)

Plug in the side length to find the perimeter:

\(\displaystyle \text{Perimeter}=6(30\sqrt3)=180\sqrt3\)