When we segment the hexagon into smaller triangles we come out with a nice looking right triangle.

We know our apothem is $\displaystyle 7\sqrt{3}$ cm, so that leaves us with a base of 7cm and a hypotenuse of 14 cm.

Looking at the picture, we can see that the hypotenuse of this triangle is also the radius of the outlying circle:

With our radius of 14 cm, we can plug into $\displaystyle Area =\pi r^2$ for circles and come up with an answer of $\displaystyle 196\pi$ $\displaystyle cm^2$

The hexagon is composed of 6 combined equilateral triangles, with 1 vertex from each equilateral joining the center point.  

Therefore, since the side length of the hexagon is $\displaystyle a+b$, and each side length of the equilateral triangle is equal, then all side lengths of the equilateral triangles must be $\displaystyle a+b$.

Two side lengths of the equilateral triangle joins to create the diagonal of the hexagon.

$\displaystyle 2(a+b)=2a+2b$

The diagonal length is $\displaystyle 2a+2b$.

The hexagon can be broken down into 6 equilateral triangles.  Each side of the equilateral triangle is equal.  Two of the combined lengths of the equilateral triangles join to form the diagonal of the hexagon.  

Therefore, the diagonal is twice the side length of the hexagon.

$\displaystyle 20\cdot 2=40$

When all the diagonals connecting opposite points of a regular hexagon are drawn in, $\displaystyle 6$ congruent equilateral triangles are created. We can also see that the length of one such diagonal is merely twice the length of a side of the hexagon.

Use the perimeter to find the length of a side of the hexagon.

$\displaystyle \text{Length of Side}=\frac{\text{Perimeter}}{6}$

$\displaystyle \text{Length of Side}=\frac{42}{6}=7$

Double the length of a side to get the length of the wanted diagonal.

$\displaystyle \text{Length of Diagonal}=2(7)=14$

When all the diagonals connecting opposite points of a regular hexagon are drawn in, $\displaystyle 6$ congruent equilateral triangles are created. We can also see that the length of one such diagonal is merely twice the length of a side of the hexagon.

Use the perimeter to find the length of a side of the hexagon.

$\displaystyle \text{Length of Side}=\frac{\text{Perimeter}}{6}$

$\displaystyle \text{Length of Side}=\frac{60}{6}=10$

Double the length of a side to get the length of the wanted diagonal.

$\displaystyle \text{Length of Diagonal}=2(10)=20$

When all the diagonals connecting opposite points of a regular hexagon are drawn in, $\displaystyle 6$ congruent equilateral triangles are created. We can also see that the length of one such diagonal is merely twice the length of a side of the hexagon.

Use the perimeter to find the length of a side of the hexagon.

$\displaystyle \text{Length of Side}=\frac{\text{Perimeter}}{6}$

$\displaystyle \text{Length of Side}=\frac{66}{6}=11$

Double the length of a side to get the length of the wanted diagonal.

$\displaystyle \text{Length of Diagonal}=2(11)=22$

When all the diagonals connecting opposite points of a regular hexagon are drawn in, $\displaystyle 6$ congruent equilateral triangles are created. We can also see that the length of one such diagonal is merely twice the length of a side of the hexagon.

Use the perimeter to find the length of a side of the hexagon.

$\displaystyle \text{Length of Side}=\frac{\text{Perimeter}}{6}$

$\displaystyle \text{Length of Side}=\frac{72}{6}=12$

Double the length of a side to get the length of the wanted diagonal.

$\displaystyle \text{Length of Diagonal}=2(12)=24$

When all the diagonals connecting opposite points of a regular hexagon are drawn in, $\displaystyle 6$ congruent equilateral triangles are created. We can also see that the length of one such diagonal is merely twice the length of a side of the hexagon.

Use the perimeter to find the length of a side of the hexagon.

$\displaystyle \text{Length of Side}=\frac{\text{Perimeter}}{6}$

$\displaystyle \text{Length of Side}=\frac{78}{6}=13$

Double the length of a side to get the length of the wanted diagonal.

$\displaystyle \text{Length of Diagonal}=2(13)=26$

When all the diagonals connecting opposite points of a regular hexagon are drawn in, $\displaystyle 6$ congruent equilateral triangles are created. We can also see that the length of one such diagonal is merely twice the length of a side of the hexagon.

Use the perimeter to find the length of a side of the hexagon.

$\displaystyle \text{Length of Side}=\frac{\text{Perimeter}}{6}$

$\displaystyle \text{Length of Side}=\frac{90}{6}=15$

Double the length of a side to get the length of the wanted diagonal.

$\displaystyle \text{Length of Diagonal}=2(15)=30$

When all the diagonals connecting opposite points of a regular hexagon are drawn in, $\displaystyle 6$ congruent equilateral triangles are created. We can also see that the length of one such diagonal is merely twice the length of a side of the hexagon.

Use the perimeter to find the length of a side of the hexagon.

$\displaystyle \text{Length of Side}=\frac{\text{Perimeter}}{6}$

$\displaystyle \text{Length of Side}=\frac{33}{6}=5.5$

Double the length of a side to get the length of the wanted diagonal.

$\displaystyle \text{Length of Diagonal}=2(5.5)=11$